Morphic Groups -
83
Morphic Groups Li 1. Introduction 2. Morphic groups 3. Direct products 4. P-groups Bibliography Morphic Groups Y. Li Brock University July, 2011
Transcript of Morphic Groups -
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Morphic Groups
Y. Li
Brock University
July, 2011
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Content
1 1. Introduction
2 2. Morphic groups2.1 Basic properties and examples2.2 Main Results
3 3. Direct products
4 4. P-groups
5 Bibliography
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]), and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])
to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]), and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]), and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]),
and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]), and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]), and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]), and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Definition and Motivation
Definition 1.1
A group G is called morphic if every endomomorphism α ofG for which Gα is normal in G satisfies G/Gα ∼= ker(α).
1 The condition for modules was introduced by Ehrlich(1976 [1])to characterize when the endomorphism ringof a module is unit regular.
2 The condition M/Mα ∼= ker(α) was studied in the con-text of rings by Nicholson and Sanchez Campos (2004[4]), and then for modules (2005 [5])(for group rings byChen, Li and Zhou, 2007).
3 A group-theoretic version of Ehrlich’s theorem is givenby Li and Nicholson (2010 [2]).
We now study this condition in the category of groups [3].
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Lemma 2.1
The following are equivalent for a group G :
(1) G is morphic.(2) If K C G is such that G/K ∼= N C G, then G/N ∼= K.
Example 2.2
Cn is morphic for each n ≥ 1, but the infinite cyclic groupC∞ is not morphic, and neither is C2 × C4.
Proof. Since Cn/ker(α) ∼= (Cn)α, groups Cn/(Cn)α andker(α) are cyclic and of the same order, and hence are isomor-phic. Hence Cn is a morphic group. However, G = C∞ = 〈a〉is not morphic because the endomorphism α : ak 7→ a2k isone-to-one but not onto, so G/Gα 6∼= 1 = ker(α).Let G = C2 × C4 = 〈a〉 × 〈b〉 and K = 〈b〉. Note thatG/K ∼=
⟨b2⟩
= N , but G/N ∼= C2 × C2 6∼= K, so C2 × C4 isnot morphic by Lemma 2.1 (2) �
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Lemma 2.1
The following are equivalent for a group G :(1) G is morphic.(2) If K C G is such that G/K ∼= N C G, then G/N ∼= K.
Example 2.2
Cn is morphic for each n ≥ 1, but the infinite cyclic groupC∞ is not morphic, and neither is C2 × C4.
Proof. Since Cn/ker(α) ∼= (Cn)α, groups Cn/(Cn)α andker(α) are cyclic and of the same order, and hence are isomor-phic. Hence Cn is a morphic group. However, G = C∞ = 〈a〉is not morphic because the endomorphism α : ak 7→ a2k isone-to-one but not onto, so G/Gα 6∼= 1 = ker(α).Let G = C2 × C4 = 〈a〉 × 〈b〉 and K = 〈b〉. Note thatG/K ∼=
⟨b2⟩
= N , but G/N ∼= C2 × C2 6∼= K, so C2 × C4 isnot morphic by Lemma 2.1 (2) �
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Lemma 2.1
The following are equivalent for a group G :(1) G is morphic.(2) If K C G is such that G/K ∼= N C G, then G/N ∼= K.
Example 2.2
Cn is morphic for each n ≥ 1, but the infinite cyclic groupC∞ is not morphic, and neither is C2 × C4.
Proof. Since Cn/ker(α) ∼= (Cn)α, groups Cn/(Cn)α andker(α) are cyclic and of the same order, and hence are isomor-phic. Hence Cn is a morphic group. However, G = C∞ = 〈a〉is not morphic because the endomorphism α : ak 7→ a2k isone-to-one but not onto, so G/Gα 6∼= 1 = ker(α).Let G = C2 × C4 = 〈a〉 × 〈b〉 and K = 〈b〉. Note thatG/K ∼=
⟨b2⟩
= N , but G/N ∼= C2 × C2 6∼= K, so C2 × C4 isnot morphic by Lemma 2.1 (2) �
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Lemma 2.1
The following are equivalent for a group G :(1) G is morphic.(2) If K C G is such that G/K ∼= N C G, then G/N ∼= K.
Example 2.2
Cn is morphic for each n ≥ 1, but the infinite cyclic groupC∞ is not morphic, and neither is C2 × C4.
Proof. Since Cn/ker(α) ∼= (Cn)α, groups Cn/(Cn)α andker(α) are cyclic and of the same order, and hence are isomor-phic. Hence Cn is a morphic group.
However, G = C∞ = 〈a〉is not morphic because the endomorphism α : ak 7→ a2k isone-to-one but not onto, so G/Gα 6∼= 1 = ker(α).Let G = C2 × C4 = 〈a〉 × 〈b〉 and K = 〈b〉. Note thatG/K ∼=
⟨b2⟩
= N , but G/N ∼= C2 × C2 6∼= K, so C2 × C4 isnot morphic by Lemma 2.1 (2) �
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Lemma 2.1
The following are equivalent for a group G :(1) G is morphic.(2) If K C G is such that G/K ∼= N C G, then G/N ∼= K.
Example 2.2
Cn is morphic for each n ≥ 1, but the infinite cyclic groupC∞ is not morphic, and neither is C2 × C4.
Proof. Since Cn/ker(α) ∼= (Cn)α, groups Cn/(Cn)α andker(α) are cyclic and of the same order, and hence are isomor-phic. Hence Cn is a morphic group. However, G = C∞ = 〈a〉is not morphic because the endomorphism α : ak 7→ a2k isone-to-one but not onto, so G/Gα 6∼= 1 = ker(α).
Let G = C2 × C4 = 〈a〉 × 〈b〉 and K = 〈b〉. Note thatG/K ∼=
⟨b2⟩
= N , but G/N ∼= C2 × C2 6∼= K, so C2 × C4 isnot morphic by Lemma 2.1 (2) �
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Lemma 2.1
The following are equivalent for a group G :(1) G is morphic.(2) If K C G is such that G/K ∼= N C G, then G/N ∼= K.
Example 2.2
Cn is morphic for each n ≥ 1, but the infinite cyclic groupC∞ is not morphic, and neither is C2 × C4.
Proof. Since Cn/ker(α) ∼= (Cn)α, groups Cn/(Cn)α andker(α) are cyclic and of the same order, and hence are isomor-phic. Hence Cn is a morphic group. However, G = C∞ = 〈a〉is not morphic because the endomorphism α : ak 7→ a2k isone-to-one but not onto, so G/Gα 6∼= 1 = ker(α).Let G = C2 × C4 = 〈a〉 × 〈b〉 and K = 〈b〉. Note thatG/K ∼=
⟨b2⟩
= N , but G/N ∼= C2 × C2 6∼= K, so C2 × C4 isnot morphic by Lemma 2.1 (2) �
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Lemma 2.1
The following are equivalent for a group G :(1) G is morphic.(2) If K C G is such that G/K ∼= N C G, then G/N ∼= K.
Example 2.2
Cn is morphic for each n ≥ 1, but the infinite cyclic groupC∞ is not morphic, and neither is C2 × C4.
Proof. Since Cn/ker(α) ∼= (Cn)α, groups Cn/(Cn)α andker(α) are cyclic and of the same order, and hence are isomor-phic. Hence Cn is a morphic group. However, G = C∞ = 〈a〉is not morphic because the endomorphism α : ak 7→ a2k isone-to-one but not onto, so G/Gα 6∼= 1 = ker(α).Let G = C2 × C4 = 〈a〉 × 〈b〉 and K = 〈b〉. Note thatG/K ∼=
⟨b2⟩
= N , but G/N ∼= C2 × C2 6∼= K, so C2 × C4 isnot morphic by Lemma 2.1 (2) �
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Corollary 2.3
Let G be morphic and let K C G. Then:(1) If K ∼= G then K = G.(2) If G/K ∼= G then K = 1.
Remark 2.4
Being morphic is a mild finiteness condition on a group.In particular, no infinite product GI can be morphic ifG 6= 1. For if G = Gi for each i, and I is well ordered,then 〈g1, g2, g3, g4, · · · 〉 7→ 〈1, g1, g2, g3, · · · 〉 is one-to-onebut not onto.
On the other hand, there are infinite morphic groups.For example, the additive group ZQ is morphic becauseend(ZQ) is a field.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Corollary 2.3
Let G be morphic and let K C G. Then:(1) If K ∼= G then K = G.(2) If G/K ∼= G then K = 1.
Remark 2.4
Being morphic is a mild finiteness condition on a group.In particular, no infinite product GI can be morphic ifG 6= 1.
For if G = Gi for each i, and I is well ordered,then 〈g1, g2, g3, g4, · · · 〉 7→ 〈1, g1, g2, g3, · · · 〉 is one-to-onebut not onto.
On the other hand, there are infinite morphic groups.For example, the additive group ZQ is morphic becauseend(ZQ) is a field.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Corollary 2.3
Let G be morphic and let K C G. Then:(1) If K ∼= G then K = G.(2) If G/K ∼= G then K = 1.
Remark 2.4
Being morphic is a mild finiteness condition on a group.In particular, no infinite product GI can be morphic ifG 6= 1. For if G = Gi for each i, and I is well ordered,
then 〈g1, g2, g3, g4, · · · 〉 7→ 〈1, g1, g2, g3, · · · 〉 is one-to-onebut not onto.
On the other hand, there are infinite morphic groups.For example, the additive group ZQ is morphic becauseend(ZQ) is a field.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Corollary 2.3
Let G be morphic and let K C G. Then:(1) If K ∼= G then K = G.(2) If G/K ∼= G then K = 1.
Remark 2.4
Being morphic is a mild finiteness condition on a group.In particular, no infinite product GI can be morphic ifG 6= 1. For if G = Gi for each i, and I is well ordered,then 〈g1, g2, g3, g4, · · · 〉 7→ 〈1, g1, g2, g3, · · · 〉 is one-to-onebut not onto.
On the other hand, there are infinite morphic groups.For example, the additive group ZQ is morphic becauseend(ZQ) is a field.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Corollary 2.3
Let G be morphic and let K C G. Then:(1) If K ∼= G then K = G.(2) If G/K ∼= G then K = 1.
Remark 2.4
Being morphic is a mild finiteness condition on a group.In particular, no infinite product GI can be morphic ifG 6= 1. For if G = Gi for each i, and I is well ordered,then 〈g1, g2, g3, g4, · · · 〉 7→ 〈1, g1, g2, g3, · · · 〉 is one-to-onebut not onto.
On the other hand, there are infinite morphic groups.For example, the additive group ZQ is morphic becauseend(ZQ) is a field.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Example 2.5
Every simple group is morphic.
Definition 2.6
Call a group G uniserial of length n if the lattice of normalsubgroups is a finite chain, that is it has the form:G = G0 ⊃ G1 ⊃ G2 ⊃ · · · ⊃ Gn = 1.
Example 2.5 says that a uniserial group of length 1 is morphic.Next we characterize when a uniserial group is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Example 2.5
Every simple group is morphic.
Definition 2.6
Call a group G uniserial of length n if the lattice of normalsubgroups is a finite chain, that is it has the form:
G = G0 ⊃ G1 ⊃ G2 ⊃ · · · ⊃ Gn = 1.
Example 2.5 says that a uniserial group of length 1 is morphic.Next we characterize when a uniserial group is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Example 2.5
Every simple group is morphic.
Definition 2.6
Call a group G uniserial of length n if the lattice of normalsubgroups is a finite chain, that is it has the form:G = G0 ⊃ G1 ⊃ G2 ⊃ · · · ⊃ Gn = 1.
Example 2.5 says that a uniserial group of length 1 is morphic.Next we characterize when a uniserial group is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Example 2.5
Every simple group is morphic.
Definition 2.6
Call a group G uniserial of length n if the lattice of normalsubgroups is a finite chain, that is it has the form:G = G0 ⊃ G1 ⊃ G2 ⊃ · · · ⊃ Gn = 1.
Example 2.5 says that a uniserial group of length 1 is morphic.Next we characterize when a uniserial group is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Basic properties and examples
Example 2.5
Every simple group is morphic.
Definition 2.6
Call a group G uniserial of length n if the lattice of normalsubgroups is a finite chain, that is it has the form:G = G0 ⊃ G1 ⊃ G2 ⊃ · · · ⊃ Gn = 1.
Example 2.5 says that a uniserial group of length 1 is morphic.Next we characterize when a uniserial group is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Theorem 2.7
Let G be a uniserial group, and let G = G0 ⊃ G1 ⊃ G2 ⊃· · · ⊃ Gn = 1 denote the lattice of normal subgroups. Thefollowing are equivalent:
(1) G is morphic.(2) If G/Gk
∼= Gn−k, k = 1, 2, . . . , n−1, then G/Gn−k ∼= Gk.
Corollary 2.8
If G is uniserial of length 2 (i.e. G has normal lattice G ⊃S ⊃ 1) then G is morphic.
Proposition 2.9
The symmetric group Sn is morphic for all n ≥ 1.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Theorem 2.7
Let G be a uniserial group, and let G = G0 ⊃ G1 ⊃ G2 ⊃· · · ⊃ Gn = 1 denote the lattice of normal subgroups. Thefollowing are equivalent:(1) G is morphic.(2) If G/Gk
∼= Gn−k, k = 1, 2, . . . , n−1, then G/Gn−k ∼= Gk.
Corollary 2.8
If G is uniserial of length 2 (i.e. G has normal lattice G ⊃S ⊃ 1) then G is morphic.
Proposition 2.9
The symmetric group Sn is morphic for all n ≥ 1.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Theorem 2.7
Let G be a uniserial group, and let G = G0 ⊃ G1 ⊃ G2 ⊃· · · ⊃ Gn = 1 denote the lattice of normal subgroups. Thefollowing are equivalent:(1) G is morphic.(2) If G/Gk
∼= Gn−k, k = 1, 2, . . . , n−1, then G/Gn−k ∼= Gk.
Corollary 2.8
If G is uniserial of length 2 (i.e. G has normal lattice G ⊃S ⊃ 1) then G is morphic.
Proposition 2.9
The symmetric group Sn is morphic for all n ≥ 1.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Theorem 2.7
Let G be a uniserial group, and let G = G0 ⊃ G1 ⊃ G2 ⊃· · · ⊃ Gn = 1 denote the lattice of normal subgroups. Thefollowing are equivalent:(1) G is morphic.(2) If G/Gk
∼= Gn−k, k = 1, 2, . . . , n−1, then G/Gn−k ∼= Gk.
Corollary 2.8
If G is uniserial of length 2 (i.e. G has normal lattice G ⊃S ⊃ 1) then G is morphic.
Proposition 2.9
The symmetric group Sn is morphic for all n ≥ 1.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Theorem 2.7
Let G be a uniserial group, and let G = G0 ⊃ G1 ⊃ G2 ⊃· · · ⊃ Gn = 1 denote the lattice of normal subgroups. Thefollowing are equivalent:(1) G is morphic.(2) If G/Gk
∼= Gn−k, k = 1, 2, . . . , n−1, then G/Gn−k ∼= Gk.
Corollary 2.8
If G is uniserial of length 2 (i.e. G has normal lattice G ⊃S ⊃ 1) then G is morphic.
Proposition 2.9
The symmetric group Sn is morphic for all n ≥ 1.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Proof of Proposition 2.9
A direct calculation shows that S1 = 1, S2 ∼= C2 × C2
and S3 ∼= D3 are morphic.
Sn is morphic if n ≥ 5 by Corollary 2.8 because thenormal lattice is Sn ⊃ An ⊃ 1.
If S4 ⊃ A4 ⊃ K4 ⊃ 1 denotes the normal subgrouplattice of S4, then S4 is morphic by Theorem 2.7 becauseS4/A4 6∼= K4 and S4/K4 6∼= A4.
Question 2.10
If G is uniserial with uniserial length 3, must G be morphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Proof of Proposition 2.9
A direct calculation shows that S1 = 1, S2 ∼= C2 × C2
and S3 ∼= D3 are morphic.
Sn is morphic if n ≥ 5 by Corollary 2.8 because thenormal lattice is Sn ⊃ An ⊃ 1.
If S4 ⊃ A4 ⊃ K4 ⊃ 1 denotes the normal subgrouplattice of S4, then S4 is morphic by Theorem 2.7 becauseS4/A4 6∼= K4 and S4/K4 6∼= A4.
Question 2.10
If G is uniserial with uniserial length 3, must G be morphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Proof of Proposition 2.9
A direct calculation shows that S1 = 1, S2 ∼= C2 × C2
and S3 ∼= D3 are morphic.
Sn is morphic if n ≥ 5 by Corollary 2.8 because thenormal lattice is Sn ⊃ An ⊃ 1.
If S4 ⊃ A4 ⊃ K4 ⊃ 1 denotes the normal subgrouplattice of S4, then S4 is morphic by Theorem 2.7 becauseS4/A4 6∼= K4 and S4/K4 6∼= A4.
Question 2.10
If G is uniserial with uniserial length 3, must G be morphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Proof of Proposition 2.9
A direct calculation shows that S1 = 1, S2 ∼= C2 × C2
and S3 ∼= D3 are morphic.
Sn is morphic if n ≥ 5 by Corollary 2.8 because thenormal lattice is Sn ⊃ An ⊃ 1.
If S4 ⊃ A4 ⊃ K4 ⊃ 1 denotes the normal subgrouplattice of S4, then S4 is morphic by Theorem 2.7 becauseS4/A4 6∼= K4 and S4/K4 6∼= A4.
Question 2.10
If G is uniserial with uniserial length 3, must G be morphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
Proof of Proposition 2.9
A direct calculation shows that S1 = 1, S2 ∼= C2 × C2
and S3 ∼= D3 are morphic.
Sn is morphic if n ≥ 5 by Corollary 2.8 because thenormal lattice is Sn ⊃ An ⊃ 1.
If S4 ⊃ A4 ⊃ K4 ⊃ 1 denotes the normal subgrouplattice of S4, then S4 is morphic by Theorem 2.7 becauseS4/A4 6∼= K4 and S4/K4 6∼= A4.
Question 2.10
If G is uniserial with uniserial length 3, must G be morphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
In [5] a leftR-moduleM is called a morphic module ifM/Mα ∼=ker(α) for every R-endomorphism α. An abelian group (writ-ten additively) is morphic as a group if and only if it is mor-phic as a Z-module.
Theorem 2.11 (5, Theorem 26)
A finitely generated abelian group is morphic if and only if itis finite and each p-primary component has the form (Cpk)n
for some n ≥ 0 and k ≥ 0.
C4×C4 is morphic, but its subgroup C2×C4 is not morphic.
Proposition 2.12
If 2 < n ∈ Z, then the dihedral group Dn is morphic if andonly if n is odd.
Proposition 2.13
No (generalized) quaternion group Q4n is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
In [5] a leftR-moduleM is called a morphic module ifM/Mα ∼=ker(α) for every R-endomorphism α. An abelian group (writ-ten additively) is morphic as a group if and only if it is mor-phic as a Z-module.
Theorem 2.11 (5, Theorem 26)
A finitely generated abelian group is morphic if and only if itis finite and each p-primary component has the form (Cpk)n
for some n ≥ 0 and k ≥ 0.
C4×C4 is morphic, but its subgroup C2×C4 is not morphic.
Proposition 2.12
If 2 < n ∈ Z, then the dihedral group Dn is morphic if andonly if n is odd.
Proposition 2.13
No (generalized) quaternion group Q4n is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
In [5] a leftR-moduleM is called a morphic module ifM/Mα ∼=ker(α) for every R-endomorphism α. An abelian group (writ-ten additively) is morphic as a group if and only if it is mor-phic as a Z-module.
Theorem 2.11 (5, Theorem 26)
A finitely generated abelian group is morphic if and only if itis finite and each p-primary component has the form (Cpk)n
for some n ≥ 0 and k ≥ 0.
C4×C4 is morphic, but its subgroup C2×C4 is not morphic.
Proposition 2.12
If 2 < n ∈ Z, then the dihedral group Dn is morphic if andonly if n is odd.
Proposition 2.13
No (generalized) quaternion group Q4n is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
In [5] a leftR-moduleM is called a morphic module ifM/Mα ∼=ker(α) for every R-endomorphism α. An abelian group (writ-ten additively) is morphic as a group if and only if it is mor-phic as a Z-module.
Theorem 2.11 (5, Theorem 26)
A finitely generated abelian group is morphic if and only if itis finite and each p-primary component has the form (Cpk)n
for some n ≥ 0 and k ≥ 0.
C4×C4 is morphic, but its subgroup C2×C4 is not morphic.
Proposition 2.12
If 2 < n ∈ Z, then the dihedral group Dn is morphic if andonly if n is odd.
Proposition 2.13
No (generalized) quaternion group Q4n is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
In [5] a leftR-moduleM is called a morphic module ifM/Mα ∼=ker(α) for every R-endomorphism α. An abelian group (writ-ten additively) is morphic as a group if and only if it is mor-phic as a Z-module.
Theorem 2.11 (5, Theorem 26)
A finitely generated abelian group is morphic if and only if itis finite and each p-primary component has the form (Cpk)n
for some n ≥ 0 and k ≥ 0.
C4×C4 is morphic, but its subgroup C2×C4 is not morphic.
Proposition 2.12
If 2 < n ∈ Z, then the dihedral group Dn is morphic if andonly if n is odd.
Proposition 2.13
No (generalized) quaternion group Q4n is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
In [5] a leftR-moduleM is called a morphic module ifM/Mα ∼=ker(α) for every R-endomorphism α. An abelian group (writ-ten additively) is morphic as a group if and only if it is mor-phic as a Z-module.
Theorem 2.11 (5, Theorem 26)
A finitely generated abelian group is morphic if and only if itis finite and each p-primary component has the form (Cpk)n
for some n ≥ 0 and k ≥ 0.
C4×C4 is morphic, but its subgroup C2×C4 is not morphic.
Proposition 2.12
If 2 < n ∈ Z, then the dihedral group Dn is morphic if andonly if n is odd.
Proposition 2.13
No (generalized) quaternion group Q4n is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
2.1 Basicpropertiesandexamples
2.2 MainResults
3. Directproducts
4. P-groups
Bibliography
Results
In [5] a leftR-moduleM is called a morphic module ifM/Mα ∼=ker(α) for every R-endomorphism α. An abelian group (writ-ten additively) is morphic as a group if and only if it is mor-phic as a Z-module.
Theorem 2.11 (5, Theorem 26)
A finitely generated abelian group is morphic if and only if itis finite and each p-primary component has the form (Cpk)n
for some n ≥ 0 and k ≥ 0.
C4×C4 is morphic, but its subgroup C2×C4 is not morphic.
Proposition 2.12
If 2 < n ∈ Z, then the dihedral group Dn is morphic if andonly if n is odd.
Proposition 2.13
No (generalized) quaternion group Q4n is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Direct products
The class of morphic groups is not closed under direct prod-ucts (C2 × C4 is not morphic). However, we do have
Proposition 3.1
If P = G×H is morphic so also are G and H.
A group is called a Hamiltonian group if it is nonabelian andevery subgroup is normal.
Example 3.2
No Hamiltonian group is morphic because it has a direct fac-tor Q8.
Question. If G and H are morphic groups, when is G×Hmorphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Direct products
The class of morphic groups is not closed under direct prod-ucts (C2 × C4 is not morphic). However, we do have
Proposition 3.1
If P = G×H is morphic so also are G and H.
A group is called a Hamiltonian group if it is nonabelian andevery subgroup is normal.
Example 3.2
No Hamiltonian group is morphic because it has a direct fac-tor Q8.
Question. If G and H are morphic groups, when is G×Hmorphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Direct products
The class of morphic groups is not closed under direct prod-ucts (C2 × C4 is not morphic). However, we do have
Proposition 3.1
If P = G×H is morphic so also are G and H.
A group is called a Hamiltonian group if it is nonabelian andevery subgroup is normal.
Example 3.2
No Hamiltonian group is morphic because it has a direct fac-tor Q8.
Question. If G and H are morphic groups, when is G×Hmorphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Direct products
The class of morphic groups is not closed under direct prod-ucts (C2 × C4 is not morphic). However, we do have
Proposition 3.1
If P = G×H is morphic so also are G and H.
A group is called a Hamiltonian group if it is nonabelian andevery subgroup is normal.
Example 3.2
No Hamiltonian group is morphic because it has a direct fac-tor Q8.
Question. If G and H are morphic groups, when is G×Hmorphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Direct products
The class of morphic groups is not closed under direct prod-ucts (C2 × C4 is not morphic). However, we do have
Proposition 3.1
If P = G×H is morphic so also are G and H.
A group is called a Hamiltonian group if it is nonabelian andevery subgroup is normal.
Example 3.2
No Hamiltonian group is morphic because it has a direct fac-tor Q8.
Question. If G and H are morphic groups, when is G×Hmorphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Direct products
The class of morphic groups is not closed under direct prod-ucts (C2 × C4 is not morphic). However, we do have
Proposition 3.1
If P = G×H is morphic so also are G and H.
A group is called a Hamiltonian group if it is nonabelian andevery subgroup is normal.
Example 3.2
No Hamiltonian group is morphic because it has a direct fac-tor Q8.
Question. If G and H are morphic groups, when is G×Hmorphic?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Question. If G is a morphic group, when is G×G morphic?
This section is devoted to partial answers to these questions.We begin with:
Proposition 3.3
Let G = G1 × G2 × · · · × Gn where the groups Gi are suchthat hom(Gi, Gj) = {θ} whenever i 6= j. Then G is morphicif and only if each Gi is morphic.
Theorem 3.4
A finite nilpotent group is morphic if and only if each Sylowsubgroup is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Question. If G is a morphic group, when is G×G morphic?
This section is devoted to partial answers to these questions.We begin with:
Proposition 3.3
Let G = G1 × G2 × · · · × Gn where the groups Gi are suchthat hom(Gi, Gj) = {θ} whenever i 6= j. Then G is morphicif and only if each Gi is morphic.
Theorem 3.4
A finite nilpotent group is morphic if and only if each Sylowsubgroup is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Question. If G is a morphic group, when is G×G morphic?
This section is devoted to partial answers to these questions.We begin with:
Proposition 3.3
Let G = G1 × G2 × · · · × Gn where the groups Gi are suchthat hom(Gi, Gj) = {θ} whenever i 6= j. Then G is morphicif and only if each Gi is morphic.
Theorem 3.4
A finite nilpotent group is morphic if and only if each Sylowsubgroup is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Question. If G is a morphic group, when is G×G morphic?
This section is devoted to partial answers to these questions.We begin with:
Proposition 3.3
Let G = G1 × G2 × · · · × Gn where the groups Gi are suchthat hom(Gi, Gj) = {θ} whenever i 6= j. Then G is morphicif and only if each Gi is morphic.
Theorem 3.4
A finite nilpotent group is morphic if and only if each Sylowsubgroup is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Question. If G is a morphic group, when is G×G morphic?
This section is devoted to partial answers to these questions.We begin with:
Proposition 3.3
Let G = G1 × G2 × · · · × Gn where the groups Gi are suchthat hom(Gi, Gj) = {θ} whenever i 6= j. Then G is morphicif and only if each Gi is morphic.
Theorem 3.4
A finite nilpotent group is morphic if and only if each Sylowsubgroup is morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Proposition 3.5
If G = S1 × S2 × · · · × Sn where each Si is simple and non-abelian, then G is strongly morphic.
Theorem 3.6
Let P = S1 × S2 × · · · × Sn × G where each Si is simpleand nonabelian, and G is morphic with the descending chaincondition (DCC) on subgroups. If no Si is a subquotient ofG, then P is morphic.
Corollary 3.7
Let P = S1 × S2 × · · · × Sn ×G where each Si is simple andnonabelian. If G is a finite, abelian, morphic group then Pis morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Proposition 3.5
If G = S1 × S2 × · · · × Sn where each Si is simple and non-abelian, then G is strongly morphic.
Theorem 3.6
Let P = S1 × S2 × · · · × Sn × G where each Si is simpleand nonabelian, and G is morphic with the descending chaincondition (DCC) on subgroups. If no Si is a subquotient ofG, then P is morphic.
Corollary 3.7
Let P = S1 × S2 × · · · × Sn ×G where each Si is simple andnonabelian. If G is a finite, abelian, morphic group then Pis morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Proposition 3.5
If G = S1 × S2 × · · · × Sn where each Si is simple and non-abelian, then G is strongly morphic.
Theorem 3.6
Let P = S1 × S2 × · · · × Sn × G where each Si is simpleand nonabelian, and G is morphic with the descending chaincondition (DCC) on subgroups. If no Si is a subquotient ofG, then P is morphic.
Corollary 3.7
Let P = S1 × S2 × · · · × Sn ×G where each Si is simple andnonabelian. If G is a finite, abelian, morphic group then Pis morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Proposition 3.5
If G = S1 × S2 × · · · × Sn where each Si is simple and non-abelian, then G is strongly morphic.
Theorem 3.6
Let P = S1 × S2 × · · · × Sn × G where each Si is simpleand nonabelian, and G is morphic with the descending chaincondition (DCC) on subgroups. If no Si is a subquotient ofG, then P is morphic.
Corollary 3.7
Let P = S1 × S2 × · · · × Sn ×G where each Si is simple andnonabelian. If G is a finite, abelian, morphic group then Pis morphic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups.
In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.
By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m.
With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:
(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Theorem 3.4 shows that deciding when a finite nilpotent groupis morphic comes down to the case of finite p-groups. In thissection, we present some results about when a p-group is mor-phic.By Theorem 2.11, a finite, abelian p-group G is morphicif and only if G ∼= (Cpk)m for some k and m. With this, thefundamental theorem gives:
Lemma 4.1
A finite abelian p-group G is morphic if and only if any twomaximal subgroups of G are isomorphic.
Part of Lemma 4.1 is true in general.
Theorem 4.2
Let G be a finite, morphic group with |G| = pn. Then:(1) All subgroups and images of G of order pn−1 are isomor-phic.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.Moreover, in this case we have:
If p = 2 then G is cyclic or elementary abelian.If p > 2 then G is either cyclic, or elementary abelian,orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.Moreover, in this case we have:
If p = 2 then G is cyclic or elementary abelian.If p > 2 then G is either cyclic, or elementary abelian,orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.
(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.Moreover, in this case we have:
If p = 2 then G is cyclic or elementary abelian.If p > 2 then G is either cyclic, or elementary abelian,orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.
Moreover, in this case we have:If p = 2 then G is cyclic or elementary abelian.If p > 2 then G is either cyclic, or elementary abelian,orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.Moreover, in this case we have:
If p = 2 then G is cyclic or elementary abelian.
If p > 2 then G is either cyclic, or elementary abelian,orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.Moreover, in this case we have:
If p = 2 then G is cyclic or elementary abelian.If p > 2 then G is either cyclic, or elementary abelian,
orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.Moreover, in this case we have:
If p = 2 then G is cyclic or elementary abelian.If p > 2 then G is either cyclic, or elementary abelian,orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.3
Let G be a group where |G| = pn, n ≥ 3, and assume thatG has a cyclic subgroup of order pn−1. Then G is morphic ifand only if it is cyclic.
Proposition 4.4
The following are equivalent for a group G with |G| = p3.(1) G is morphic.(2) Either G is cyclic, or elementary abelian, or nonabelianwith exp(G) = p.Moreover, in this case we have:
If p = 2 then G is cyclic or elementary abelian.If p > 2 then G is either cyclic, or elementary abelian,orG = 〈a, b, c | ap = bp = cp = 1, [a, b] = c, ac = ca, bc = cb〉 .
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.5
If p is a prime, every morphic group of order p4 or p5 isabelian.
Question If p is a prime, is every morphic group of orderpn with n > 3 abelian?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.5
If p is a prime, every morphic group of order p4 or p5 isabelian.
Question If p is a prime, is every morphic group of orderpn with n > 3 abelian?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.5
If p is a prime, every morphic group of order p4 or p5 isabelian.
Question If p is a prime, is every morphic group of orderpn with n > 3 abelian?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
P-groups
Proposition 4.5
If p is a prime, every morphic group of order p4 or p5 isabelian.
Question If p is a prime, is every morphic group of orderpn with n > 3 abelian?
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Bibliography
G. Ehrlich, Units and one-sided units in regular rings,Trans. Amer. Math. Soc. 216 (1976), 81-90.
Y. Li and W.K. Nicholson, Ehrlich’s Theorem for Groups,Bull. Austral. Math. Soc. 81 (2010), 304-309.
Y. Li, W.K. Nicholson and L. Zan, Morphic Groups, J.Pure Appl. Algebra 214 (2010), 1827-1834.
W.K. Nicholson and E. Sanchez Campos, Rings with thedual of the isomorphism theorem, J. Algebra 33 (2004),391-406.
W.K. Nicholson and E. Sanchez Campos, Morphic mod-ules, Comm. in Algebra 33 (2005), 2629-2647.
MorphicGroups
Li
1.Introduction
2. Morphicgroups
3. Directproducts
4. P-groups
Bibliography
Thank you!
Author: Yuanlin LiAddress: Department of Mathematics
Brock UniversityCanada
Email: [email protected]
