Math. Z. 129,65- 73 (1972)@ by Springer-Verlag 1972

Isomorphism of Modular Group Algebras

lnder Bir S. Passi and Sudarshan K. Sehgal

1. Introduction

For a group G, let Mj(G) be the i-th term in its Brauer-Jennings-Zassenhaus series which is defined inductively by M1(G)= G, Mj(G)=(G, Mj-l(G») M(ilp)(G)P for i>2 where (i/p) is the least integer >i/p and(G, Mi-l (G») denotes the subgroup generated by all commutators(x, y) = x-I y-l Xy, XEG, YEMj -1 (G). It is known ([1,5,6, 11, 13]) thatif k is a field of characteristic p > 0 and L1k(G) is the augmentation idealof the group algebra k (G), then

Mj(G)= {gEGlg-l EL1~(G)}.

It is also known [6] thatMn(G)= fl Grj

j~~n

where Gi is the i-th term in the lower central series

G =G > G >...> G.>...1= 2= = 1= ofG.

Our first main result is that if G and H are two groups with iso-morphic group algebras over the field Zp of p elements for a prime p,then they have isomorphic M-series for that prime. We exploit the factthat the natural monomorphism

0 ~ Mj (G)/Mj + 1(G) ~ L1~ (G)/L1~+I(G)p p

splits over Zp to deduce that, in fact, the subquotients Mj (G)/Mj + 2 (G)are also isomorphism invariants of the group algebra Zp(G) of G withcoefficients in Zp.

We next consider the case when the coefficients are in the ring Zphof integers mod ph, p prime. Corresponding to the M -series, we havethe series [6]

Mn,h,p(G)= nGrj

where the product is taken over all i, j such that i> n or i pu- h+1)> n.We denote by Dj,h,p(G) the i-th dimension subgroup of G with coef-ficients in the ring Z...h i.e. Di h P (G)={glg-lEL1~ h(G)} where Az h(G)

v' , , p p

is the augmentation ideal of Zph(G). By a result of Moran [8] Mj,h,p(G)=5 Math. Z., Bd. 129

66 I. B.S. Passi and S.K. Sehgal:

Dj,h,p(G) for i<p. We show that the counterpart of the splitting men-tioned in the previous paragraph is available for the natural embedding

o~ Gz/Gzn DJ(G, R)~Lt~(G)/Lt~(G)

where R is the ring of integers mod n, n odd, or R is the ring of integersand G/Gz is of odd exponent. DJ (G,R) stands for the 3rd dimensionsubgroup of Gover R. Thus, for example, we show that whenever fortwo groups G and H .

Zp" (G) '" Zp"(H), p prime=l=2,

then G/M3,n,p(G)'" H/M3,n,p(H) and moreover if, in addition, G and Hare each of exponent pn and class 2, then G'" H.

2. Notations

Let G be a group, R a commutative ring with unity. We adopt thefollowing notation:

R (G)= the group ring of G with coefficients in R.

AR(G)=the augmentation ideal of R(G).Qn,R (G) = A~ (G)JLt~+l(G).

Lt~](G)=LtR(G) for i= 1 and Lt~l(G)=[LtR(G), Lt~-lJ(G)]

the additive subgroup generated by all elements of the type

afJ-fJa, aELtR(G), fJELt~-l](G) if i>1.

LtR(G, N)=Kemel of the natural homomorphism R(G)~R(G/N),where N is a normal subgroup of G.

G=G1>GZ>...>Gj>'" denotes the lower central series of G.Zn= the ring of integers mod n.

3. M-Series

Lemma 1. Lt~J(G)+ Ltk+1(G)=LtR(G, Gj)+ Ltk+l(G)for all i> 1.

Proof Induct on i or see [2, Theorem 6].

Following Zassenhaus [13] we define a series of ideals Ln,h,P(G)

of Z ph(G) by setting

L (G)= '" ph-l(Lt[i])p.i+Lt[nJ(G)+Ltn+l(G)n, h, p L...

where A = AZph' the summation is over all i, j such that i < n, i [l> nand by SIJ'"for a subset S of Zph(G) we understand the Zph-subrnoduleof Zph(G) generated by spy,SES.

Isomorphism of Modular Group Algebras 67

Lemma 2. Mn.h.p(G)fDn+l.h.P(G)() Mn.h.p(G)'" Ln.h.p(G)fAn+1(G) forall n> L

Proof By definition Mn.h.p(G)= n+ G(ipU-h+l) ~n

For gEGi, we have g-1EA(GJ and by Lemma 1 g-1=a+p forsome aEA[iJ(G), PEAi+l(G).

If for some integer j, ipU-h+l)+ >n, then U-h+ 1)+ =0 would meani>n, gpiEGn and so gpi-1EA[nJ(G)+An+l(G). If j-h+1>0, thenconsider

pi .

gpi -1 = L (pi

)(g-1Y.r=1 r

Now ph divides (~) for r<pj-h+l and for r> pi-h+l we have i r>

i pi-h+l > n. Hence, module Ln.h.p(G)

gpi - 1= (. pi

) ( - p"J- h + I

pi-h+l g r

- a ph-l(g-1)Pj-h+1

=a ph-l(a+ pr-h+1

=0.

where a is an integer [9]

The equationx y-1 =(x-1)+(y-1)+(x-1)(y-1)

allows us to conclude that

(*)

mEMn.h.p(G) ~ m-1ELn.h.P(G).

Define a map (): Mn.h.p(G)~Ln.h.P(G)fAn+l(G)

m-+m-1 +An+l(G).

It follows from (*) that () is a homomorphism and clearly the kernel

of () is Dn+l.h.p(G). It remains to prove that () is an epimorphism. Inview of Lemma 1, we have only to verify that if gEGi and ipi>n, thenph-l (g - 1)pi has a preimage.

Now

(

...i+h-l

)gpi+h-I-l= J! pi (g-1)pjmodAn+l(G)

= a ph-l (g - l)pJ where a is an integer [9].

Since there exists an integer b such that a b = 1(mod ph), we have

b(gpj+h-I -1)= ph-l(g -lr mod An+l(G).5*

68 I. B.S. Passi and S.K. Sehgal:

Consequently +h 1 .gbpi - _1=ph-l(g-l)pJmodLtn+l(G)

and () is onto as gpi+h-I EGr+h-l c Mn.h.p(G).

Theorem 3. Let (): Z p (G) -+ Z p (H) be an isomorphism. Then

(i) Mi (G)/Mi+ 1(G)'" Mi(H)/Mi+l (H),

(ii) Mi(G)/Mi+2 (G)'" Mi(H)/Mi+2 (H)

for all i> I, where M1(G)=G=>M2(G)=>...=>Mi(G)=>... is the M-serieswith respect to the prime p.

Proof We remark first that one can assume without loss of generalitythat () is normalized in the sense that the sum of the coefficients of B(g)is 1 for all gE G. Thus the expression on the right hand side of the iso-morphism in Lemma 2 is a ring invariant and (i) follows from the factthat Di.l.p(G)=Mi(G)=Mi.l.p(G).

Consider the embedding

Mi(G)/Mi+l (G)-+ Lli(G)/Lli+1(G), Lt=Llzp'

m+Mi+l (G)-+ m-I +Lli+l(G), mEMi(G).

Since Lli(G)/Lli+l(G) is a vector space over Zp, the embedding splitsover Z p and we have

Lli(G)/LIi+1(G) =LI (G, Mi)+ Lli+l(G)/Lli+l (G) Ef)Ki (G)/LIi+1(G), (**)

where Ki(G) is a subspace of Zp(G). Moreover, since Lli+l(G)cKi(G)cLli(G), we can conclude that Ki(G) is an ideal of Zp(G). We claimthat Mi(G)/Mi+2 (G) is isomorphic to the group of units Mi(G)+Ki+l (G)/Ki+l (G) of Zp(G)/Ki+l (G). Define the map

A: Mi(G)-+ Mi(G)+ Ki+l (G)/Ki+l (G)

m-+m+Ki+l(G).

Clearly A is an epimorphism and Mi+2(G) is contained in the kernel of A.Suppose A(m)=I+Ki+l(G) i.e. m-IEKi+l(G). Then m-lELli+l(G)and so mEMi+l(G). But because LI(G,Mi+l)nKi+l(G)CLli+2(G), weconclude that mEMi+2(G) proving the isomorphism.

By Lemma 4(a) below, (**) yields the splitting

Lli(H)/Lt i+ 1(H)= LI(H, Mi (H») + Lli+l (H)/ Lli+l (H)Ef)()(Ki(G»)/Lli+l(H).

given by

Therefore

Mi(H)/Mi+2 (H)'" Mi(H)+ ()(Ki+l (G»)/B(Ki+l (G»)

'" Mi(G)+ Ki+l (G)/Ki+l (G) (Lemma 4(b) below)

'" Mi (G)/Mi+ 2 (G).

Isomorphism of Modular Group Algebras 69

Lemma 4. Under the assumptions of Theorem 3, we have

(a) 0(.1 (G, Mn(G»+An+l(G»)=A(H, Mn(H»)+An+l(H)

(b) O(Mn(G)+ Kn+l (G»)=Mn(H)+O(Kn+l (G»)

where Kn+l (G) is an ideal of Zp(G) yielding a splitting of An(G)/An+l(G)as in (**).

Proof (a) Every element x of A(G, Mn(G»)+An+l(G) can be written asx= m-l +CX,mEMn(G), cxEAn+l(G), since Mn(G)c 1+An(G). As in theproof of Lemma 2, m-lELn,l,p(G) and therefore B(m-l)ELn,l,p(H).Also O(cx)ELn,l,p(H). Hence O(x)ELn,l,p(H). By Lemma 2 again O(x)=h-l +y, hE Mn(H), YEAn+l(H) and we are done.

(b) Let mE Mn(G). Then O(m-l)=h-l +y for some hEMn(H) andYEAn+l(H). As O(Kn+l (G») yields a splitting, y=u-l +v for someuEMn+l (H), vEO(Kn+l (G»).

Therefore,

O(m-l)=h-l +u-l +v

=hu-l+w where w=v-(h-l)(u-l).

As (h-l)(u-l)EAn+2 (H)=O(An+2 (G»)cO(Kn+l (G»), wEO(Kn+l (G»).Since hUE Mn(H), the proof of the Lemma is complete.

Corollary 5. Let Zp(G)'" Zp(H). Then Mj(G)=(I) if and only ifMj(H)=(l).

Proof Suppose Mj(G)=(I). Then, by Theorem 3, Mj(H)=Mj+j(H)for all j > o. Mj (G) = (1) implies that G is a nilpotent p-group of boundedexponent. Thus we can conclude [4, Theorem E] that n A'(G)=(O),where A =Azp and, therefore, nA' (H) = (0). It follows that Mj(H)=(l).,The converse follows by symmetry.

Related to Corollary 5 is the following remark:

Remark 6. Let Zp(G)"" Zp(H). Then n Mj(G)=(I) if and only ifn Mj(H) = (1). j

j

Proof Suppose nMj(G)=(I). Then G is residually nilpotent p-groupj

of bounded exponent. Therefore, nA~p(G)=(O) ([4, Theorem E] andj

[7, Lemma 5.2]). Thus nA~ (H)=(O) and, therefore, nMj(H)=(l).. p .. I

The converse follows by symmetry.Coronary 5 and Theorem 3 give the following:

Corollary 7. Suppose G has M-length <2 i.e. M3(G)=(1). ThenZp(G)'" Zp(H) implies G'" H.

70 I. B.S. Passi and S. K. Sehgal:

The next result has been proved by Dieckmann [3] for finite p-groupsof class 2, p =1=2.

Corollary 8. Let G and H be two groups such that

(i) Zp(G)'" Zp(H)(ii) G is of class < p and exponent pn.

Then H is of exponent pn.

Proof. We observe that Mp..(G)= n Gf=(1)and that Mk(G)=I=(l)ipi~p"

for k<~. It follows by Corollary 5 that Mp..(H)=(l) and Mk(H)=I=(I)for k < pn.Hence H is of exponent pn.

4. Groups of Class Two

Let R be a commutative ring with unity, G a group. Consider thenatural epimorphism

a: R(G)~R(GjG2)

a{ L rg)=L rg whereg=gG2.reRgeG

a induces an epimorphism a: Q2.R(G)~Q2.R(GjG2) whose kernel isevidently LfR(G, G2)+Lfj(G)jLfj(G).

Consider the map

-r: GjG2 ~Q2.R(GjG2) given by -r(xG2)=(x-l)2 +Lfj(G).

T is well-defined. For, if YEG2, then

(x y-l)2 ={x-l + y-l +(x-l)(Y-l»)2

=(x-l)2 mod Lfj(G), since G2c 1+Lfi(G).

Extend -rby linearity to R(GjG2). We assert that T vanishes on Lfi(G/G2).For, let x,y, ZEG. Then, writing x=xG2 etc., we have

-r{(x-l)(Y-l){z-l»)=(x y z-lf -(xy-l)2 -(x z-I)2 -(y z-I)2

+(x-l)2 +(y-l)2 +(z -1)2 + Lfi(G)

=(x-l + y-l +z-I)2 -(x-l + y_1)2

-(x-l +z-lf-(y-l+z-1f

+(x-l)2 +(y-lf +(z-lf + Lfi(G)=0.

Thus -rinduces an R-homomorphism

T: R(GjG2)jLfi(GjG2)~Q2.R(G).

Isomorphism of Modular Group Algebras 71

Restricting to Q2,R(G/G2) we obtain a homomorphism

-r: Q2,R(G/G2)-+Q2,R(G).

Theorem 9. IX0 -r= 21 where I is the identity homomorphism.

Proof Let x=xG2, y=yG2 be elements of G/G2.

-r(x-l)(y-l»)=(x y-l)2 -(x-l)2 -(y-lf + L1~(G)

=(x-l)(y-l)+(y-l)(x-l)+L1MG).

Hence IX0 -r(x-l)(Y-l)+ L1~(G/G2»)=2(x-l)(Y-l)+ L1~(G/G2).

As the elements (x-l)(Y-1)+L1~(G/G2) generate Q2.R(G/G2)as anR-module, the result follows.

Corollary10. If division by 2 is uniquely defined in Q2.R(G/G2), thenthe exact sequence

O-+L1(G, G2)+L1~(G)/L1~(G)-+Q2.R(G)~ Q2,R(G/G2)-+ 0splits.

Proof Define -r': Q2.R(G/G2)-+Q2.R(G) by

-r'(z)=-r(y) where 2y=z, y, ZEQ2.R(G/G2).Then we have IX0 -r'= I.

Let us examine the case when R=Zn, the ring of integers mod n,n odd, or Z the ring of integers. The case n = 0 in the following discussioncorresponds to Z as the ring of coefficients. For every group G we havethe exact sequence

O-+G2/G2 n D3(G, Zn)~Q2.Zn(G)~Q2.Zn(G/G2)-+O

where i is the natural map induced by

(4.1)

i: G2-+Q2.zJG)

x-+ x-I +L1t(G)

and Dr(G, Zn) denotes the r-th dimension subgroup of Gover Zn' Wealso have the natural isomorphism

G/G2 Gn", L1zn(G)/L1t(G)

induced by x -+ x-I + L1t (G),XEG.Suppose that the exact sequence (4.1) splits over Zn' Then we have a

Zn-submodule K of L1iJG) such that K:::>A~JG) and therefore is, infact, an ideal of Zn(G) and

(4.2)

Q2.Zn(G)= AZn(G, G2)+ At (G)/L1t (G) EBK/At (G). (4.3)

72 L B.S. Passi and S. K. Sehgal:

Consider the ring S = Zn (G)/K. U = 1+ j Zn (G)/K is a group of units of Sand we have a homomorphism A: G~ U given by A(x)=x+K.

Let OEj zJG). By (4.2)

o=g-1+b2, ozEjiJG)

=g-l +g2 -1 +k, gzEGz, kEK (by (4.3»)

=gg2-1 mod K.

Hence A(g g2)= 1+ 0 + K. This proves that A is an epimorphism. LetxEKer A. Then x-1EKcjt(G). Therefore, xEG2 Gn, say x= y Z,YEG2,zEGn. Now

Y z-l = y-1 +z-l +(y-1)(z-1).

The last term on the right hand side is in K, since K ~ jiJG). For gEG,

gn-l =n(g-l)+ (~) (g-1)2 modjt(G)

=0 since n is odd or O.

Thus GncD3(G, Zn)'

Hence z-lEK. Consequently y-1EK. As YEG2, (4.3) implies thaty-lEjiJG) i.e. YED3(G,Zn) and so xED3(G,Zn)' We have provedthe following

Theorem 11. If n is an odd integer or 0 and the sequence

0~G2/G2(\D3(G, Zn)~Q2.z,,(G)~Q2.Z,,(G/G2)~0

splits over Zn, thenG/D3 (G, Zn)'" 1+ jzJG)/K

where K/jiJG) is a complement of 1m i in Q2.Z,,(G).

If n is an odd integer or G is a group of odd exponent, thenCorollary 10 yields that the hypothesis of Theorem 11 is satisfied. Thecase n = 0, i.e. the integral case, for finite groups is due to Sandling([10, 11]). By Lemma 1 jz,,(G, G2)+jiJG) can be characterized ringtheoretically. Thus if, for some group H, Zn(G)'" Zn(H), then the sequencecorresponding to (4.2) for H would also split. We thus have

Theorem 12. If n is an odd integer or G is a group of odd exponent,and Zn(G)'" Zn(H) for some group H, then

G/D3 (G, Zn)'" H/D3 (H, Zn)'

Corollary 13. If G and Hare p-groups of exponent pn, p=t2 and class 2

with Zpn(G)'" Zpn(H), thenG'" H.

Isomorphism of Modular Group Algebras 73

Proof By Theorem 12

G/D3(G, Zp")'" H/D3 (H, Zpn).Now

D3(G, Zpn)= GpnGr G3 [8]

=(1) (since G is of exponent pn and class 2)

and similarly D3 (H, Zpn)= (1). Hence G'" H.

Remark 14. It has been proved in [12] that for finite groups G and Hwith IGI=IHI=pmb, (P,b)=1 the isomorphism Zp2m+l(G)"'Zp2m+dH)implies the isomorphism Z;( G)= Z;(H) where Z; is the ring of p-adicintegers. Hence it follows [12] that if b = 1 and Z p2m+1(G)'" Zp2m+l(H)then G/G3 '" H/H3' The last two results replace the modulus p2m+l bya much smaller power of p.

References

1. Bovdi, A. A.: Dimension subgroups. Proceedings of the Riga Seminar on Algebra5-7(1969).

2. Cohn, P.M.: Generalization ofa theorem of Magnus. Proc. London Math. Soc. (3) 2,297-310 (1952).

3. Dieckmann, E.M.: Isomorphism of group algebras of p-groups. Ph.D. dissertation,Washington University, St. Louis, Missouri, 1967.

4. Hartley, B.: The residual nilpotence of wreath products. Proc. London Math. Soc. (3)20,365-392 (1970).

5. Jennings, S.A.: The structure of the group ring of a p-group over a modular field.Trans. Amer. Math. Soc. SO,175-185 (1941).

6. Lazard, M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. Ecole NormSup. (3)71, 101-190(1954).

7. Mital,J.N.: On residual nilpotence. J. London Math. Soc. (2), 2, 337-345 (1970).8. Moran, S.: Dimension subgroups modulo n. Proc. Cambridge Philos. Soc. 68, 579-582

(1970).9. Passi, I.B.S.: Polynomial maps on groups. J. Algebra 9,121-151 (1968).

10. Sandling, R: Note on the integral group ring problem. Math. Z. 124,255-258 (1972).11. Sandling, R.: The modular group rings of p-groups. Thesis, University of Chicago, 1969.12. Sehgal, S.K.: Isomorphism of p-adic group rings. J. Number Theory 2, 50<>-509(1970).13. Zassenhaus, H.: Ein Verfahren jeder endlichen p-Gruppe einen Lie-Ring mil der

Charakteristik p zuzuordnen. Abh. Math. Sem. Univ. Hamburg 13, 200-207 (1940).

Dr. I. B.S. PassiDepartment of MathematicsUniversity of KurukshetraKurukshetraIndia

Prof. S. K. SehgalMathematisches Institut der UniversiHitD-6900 HeidelbergFederal Republic of Germany

(Received May 2,1972)