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Commutative Group Algebras COMMUTATIVE GROUP ALGEBRAS BY WARREN MAY Let R be a commutative ring with identity. If G is a group, then one can form the group algebra of G over R which we shall denote by R[G]. One can then ask the basic question, how much information about the group G can be deduced from the F-algebra R[G]7 A slightly different formulation is to assume that G and //are groups with R[G] isomorphic to R[H] as /^-algebras and to ask what relations exist between G and H. A discussion of the problem with references can be found in Curtis-Reiner [2, p. 262]. There they prove the early result of Higman [4] that if R is the ring of integers and the groups are finite abelian, then isomorphism of the group algebras implies isomorphism of the groups. Later work of Perlis-Walker [8] and Deskins [3] takes up the problem over fields. One result is that if we take R to be the rational numbers, then the group algebras determine finite abelian groups. There are difficulties in trying to apply the methods of group representations to other situations, particularly if the characteristic of R is finite. However, if the group algebra is commutative, i.e., the group is abelian, then in case R has charac- teristic p, one has the Frobenius endomorphism of R[G]. In this paper we show that systematic application of this mapping enables one to deduce fairly broad con- clusions about abelian groups with isomorphic group algebras over commutative rings. In fact the group modulo its torsion subgroup is completely determined. To deduce conclusions about a /^-primary component of its torsion subgroup however, we must require that R behaves sufficiently well with respect to the prime p. More precisely, one requires that p is not invertible in R. Under this condition, the maximal divisible subgroup and the Ulm invariants of the /^-primary component are determined. A complete statement of results is contained in the theorem in the last section. In the middle two sections we work over fields of various types and then generalize to arbitrary commutative rings later. 1. Preliminaries. It is worthwhile to prove several standard propositions on group algebras since they will be used repeatedly in the succeeding sections. Let R be a fixed commutative ring with identity. Then the formation of group algebras over R gives a functor from the category of groups to the category of P-algebras. If /: G -*■H is a homomorphism of groups, then /induces an P-algebra homo- morphism/': R[G] ->■R[H] byf'(1 rgg) = Jirgf{g). If we identify R with the group algebra of a fixed trivial group, then the homomorphism of G onto this trivial Received by the editors February 16, 1967 and, in revised form, November 27, 1967. 139 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 140 WARREN MAY [February group induces an /^-algebra homomorphism of R[G] onto R called the augmenta- tion map and denoted i: R[G] -> R. Then i is given by j'(2 rgg) = '2,i'g- The kernel of i is called the augmentation ideal of Ä[G]. Our first proposition says that if we have an arbitrary /?-algebra map of R[G] into R then one can change the group algebra by an automorphism so that the given map becomes the augmentation map. A reference in the literature for this result is Losey [6, Theorem 2.3]. Proposition 1. Let G be a group and i the augmentation map on R[G]. Suppose T is an R-algebra such that T^R[G] as R-algebras. Assume we are given an R- algebra map V: R. Then there exists an R-algebra isomorphism y. T —s*R[G] such that i ° <p= i'. Proof. Equivalently, given z": R[G]^R, we may ask for an automorphism y of R[G] such that i°(p = i'. For g e G, put rg = i'(g). Define <p:G —>R[G] by <p(g)= rgg. Then <pis a homomorphism and so by the mapping property of group algebras, <pextends to an Ä-algebra map <p:R[G]-+R[G]. It is easy to show <p has an inverse (since (rg)~1 = rg-i) and that i ° <p= i'. H Corollary 1. Let G± and G2 be groups and M1 and M2 the augmentation ideals of /?[GJ and R[G2] respectively. If RIG^^RIG^ as R-algebras, then we may choose an isomorphism <psuch that <p(M^)= M2. At this point the author would like to thank the referee for his helpful remarks and for suggesting the next two corollaries. Suppose that <p\ R[Gx] -> R[G2] is an isomorphism commuting with the augmentation maps, and that A is an 7?-module. Let Gj act on A trivially and let G2 act on A by the action induced by <p.It then follows that G2 acts trivially on A. Corollary 2. Let Gx and G2 be groups and A an R-module on which we let Gj and G2 act trivially. Assume R[Gy\^R[G2] as R-algebras. Then HXG^A)^ Hn(G2, A) and Hn(Gx, A) = Hn(G2, A) for all 0 (where homology and cohomology are taken over R[Gi] or R[G2]). Applying this corollary to the integers and recalling that H^G, Z) is the abeliani- zation of G, we obtain Corollary 3. Two groups having isomorphic group algebras over the integers have isomorphic abelianizations. Hence, if the two groups are abelian, then they are isomorphic. The next proposition is stated in a slightly more general form than one would normally use, but this will simplify later applications. For comparisons, see Connell [1, Proposition 1] and Gruenberg [5, Lemma 2]. We shall use the letter e to denote identity elements of groups. Presumably no confusion will arise from using it to denote the identity elements in different groups simultaneously. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1969] COMMUTATIVE GROUP ALGEBRAS 141 Proposition 2. Let G be a group, S a subset of G, and H the normal subgroup of G generated by S. Let I be the two-sided ideal of R[G] generated by the set of elements {s —e\seS}. Let j.R^-R be an injective ring homomorphism and suppose f.G^y GjH is the natural map. Then j and f induce a ring homomorphism <p:R[G] R[G/H] by <p(2r9g) = lj(rg)f(g). The kernel oj> is I. Proof. For seS it is clear that <p(s-e)=0, hence /£(kernel <p).Suppose that gi-e and g2-e are both in /. Then g1g2-e = (g1-e)(g2-e) + (gi-e) + (g2-e) is in /. Also if (g-e) e I, then g'1 —e= —g~1(g—e) is in /. Finally if (g —e) e / and beG, then bgb'1 -e = b(g-e)b~l is in /. It follows now that {h-e | Aef/}g/. Choose {gj to be a complete family of coset representatives of H in G. Let a e R[G], say a = 2i,h rUhgih where he H. Then f.ft i \ ft Ii \ ft I Hence y(a) = 0 implies thaty'(2ft riih) = 0 for all / and so ri ft= 0 for all r. It follows that (—fi,h)=fi.t and so 2a ri,hh = 2.h*e i"i,hih— e) which is contained in /. Therefore « = 2igi(2/i ''i.hA) is contained in /. Hence (kernel <p)s /. | Corollary 4. Tfte augmentation ideal of R[G] is generated by {g—e \ g e G}. Corollary 5. Let H be a subgroup of G and M the augmentation ideal of R[G]. Then, regarding R[H] as a subalgebra of R[G], the augmentation ideal of R[H] is M n R[H], IfH is normal and I is the two-sidedideal of R[G] generated by M n R[H], then R[G]/I^R[G/H]. Proof. It is clear that M n R[H] is the augmentation ideal of R[H]. Since {h —e I h e H} generates M n R[H] as an ideal in R[H], the same set generates / as an ideal in R[G]. The claim now follows. | 2. The torsion-free case. By a localized polynomial ring we shall mean a poly- nomial ring over a commutative ring with identity which is localized at the multi- plicative monoid of monomials. The first two lemmas we prove can be traced back to Higman [4, Theorem 12 and 13]. Lemma 1. Let F be a field and G an abelian group. Then F[G] has nontrivial zero divisors if and only if G is not torsion-free. Proof. Any two elements of F[G] are contained in a subalgebra of form F[GX] where Gx is some finitely generated subgroup of G. Then G torsion-free implies Gi is a free abelian group, hence F\G-t\ is isomorphic to a localized polynomial ring over F in a finite number of variables. Therefore P[Gi] is a domain, hence F[G] must be a domain. If G is not torsion-free, then G has a nontrivial finite subgroup, call it K. Put a=2*e/r*- Then if | AT| =«, we see a2 = na, and so a(a—«) = 0. Hence both a and a —n nonzero implies PfG] has nontrivial zero divisors. | License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 142 WARREN MAY [February Lemma 2.
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