Iso Morph Ism

8/6/2019 Iso Morph Ism

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Definition

An isomorphism G G

is a homomorphism that isone to one and onto G.

The usual notation isG G.


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Theorem Let S be any collection of groups, and

define G G in S if there exists an

isomorphism G G. Then isan equivalence relation.


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Proof Exercise


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How to Show That GroupsAre Isomorphic

Define the function that gives theisomorphism of G with G. Now this means

that we have to describe, in some fashion,what (x) is to be in G for everyx G.

Show that is a one to one function.

Show that is onto G.

Show that is a homomorphism.


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Example 1Show that (R , +) (R+ , .)

Solution

1. Define R R+ is given by (x) = ex, forall x R.

2. Show that is a one to one function.

3. Show that is onto R+

.4. Show that (x + y) = (x) (y).


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TheoremAny infinite cyclic group

G is isomorphic to thegroup Z of integers

under addition.


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Proof We suppose that G has a generator

a and use multiplicative notation

for the operation in G. ThusG = { an | n Z }.

Remember that the elementsan G are all distinct, that is,an am if n m.


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Proof (Cont.)1. Define G Z is given by

(an) = n, for all an G.

2. Show that is a one to one function.

3. Show that

is onto Z.

4. Show that (anam) = (an) + (am).


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When we say, There are precisely m

groups of order n, up to isomorphism, wemean that there exist m groups G1, G2, ...,Gm of order n, no two of which areisomorphic, and that every group of order n

is isomorphic to one of them.

For example, we have seen that any twogroups of order 3 are isomorphic.

We express this by saying that there is onlyone group of order 3 up to isomorphism.


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Example 2 There is only one group of order 1, one of

order 2, and one of order 3 up to

isomorphism. There are exactly two different groups of

order 4 up to isomorphism, the group Z4 andthe Klein 4-group V.

There are at least two different groups oforder 6 up to isomorphism, namely

Z6 and S3.


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How to Show That Groups

Are Not Isomorphic This would mean that there is no one-to-one

function from G onto G with the property(ab) = (a) (b).

In general, it is clearly not feasible to tryevery possible one-to-one function to find outwhether it has the above property, except in

the case where there are no one-to-onefunctions. This is the case, for example, if Gand G are of finite order and have differentnumbers of elements.


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Example 3Z4 and S6 are notisomorphic.

Since there is no one-to-onefunction from Z4 onto S6.


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For two groups of infinite order, it is notalways clear whether there are any one-to-

one onto functions.

For example, you may think that Q has moreelements than Z, but your instructor canshow you in five minutes that there are lostof one-to-one functions from Z onto Q.

However, it is true that R has toomany elements to be put into a one-to-one correspondence with Z.


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Example 4(Z , +) is not isomorphic with (R , +).

Since there is no one-to-one functionfrom Z onto R.


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In the event that there are one-to-onemappings of G onto G, we usually show

that the groups are not isomorphic (ifthis is the case) by showing that onegroup has some structural property thatthe other does not possess.

A structural property of a group is onethat must be shared by any isomorphicgroup.

It must not depend on the names orsome other nonstructural characteristicsof the elements.


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The following are examples of somestructural properties and somenonstructural properties of groups.

Possible StructuralProperties

1. The group is cyclic.

2. The group is abelian.3. The group has order n.4. The group is finite.5. The group has exactly

two elements of order 5.

6. The equation has asolution for eachelement a in the group.

Possible NonstructuralProperties

1. The group contains 5.2.

All elements of the groupare numbers.3. The group operation is

called composition.4. The elements of the group

are permutations.5. The group operation is

denoted by juxtaposition.6. The group contains no

matrices.

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