Some isomorphism results on commutative group algebras

Throughout the present paper, suppose FG is the group algebra of an abelian group G written multiplicatively (and possibly mixed) over a field F of positive characteristic p. As usual, FG denotes the group algebra of G over F with unit group U (FG) and normalized unit group V(FG); note that the direct decomposition U(FG) = V(FG) x F * holds where F ^∗ is the multiplicative group of F and thus the study of U(FG) reduces to the study of V (F G). Moreover, let S(FG) = V _p (F G) and let G _p be the p-primary components of V (FG) and G, respectively. All other notions and notations are standard and follow essentially those from the reference list at the end of the article. Nevertheless, we will give below some supplementary terminology and concepts.

Two central subjects in commutative group algebras theory have been played a major role. First of all, this is the isomorphism problem for commutative modular group algebras which is still unresolved and seems insurmountable in full generality at this stage. It states as follows:

Isomorphism Conjecture . Suppose G is a p -mixed group. If H is any group and FG = FH as F-algebras, then H = G.

This question was settled by many authors for various classes of abelian groups; for example the interested reader can see [1–9] together with the complete references in [10–12]. Here we shall confirm once again its truthfulness provided G is splitting whose maximal torsion subgroup G t is a totally projective (reduced or not) p -group (compare also with the original source [1]). Note that when the torsion subgroup G _t is a torsion-complete p -group, the readers may see [3], [5] and [8].

All of this is subsumed by the second challenging topic. Specifically, we state the following yet left-open.

Direct Factor Conjecture . Let G be a p-mixed group and let F be perfect. Then V ( F G ) /G is simply presented and, in particular, G is a direct factor V ( FG ) with simply presented complement.

The structure of V (FG) has a great deal of an intensively study by many authors; for instance the interested reader can see [1–9] along with the complete bibliography in [10–12].

Here we will investigate how G is situated inside V (F G) in some special cases, mainly motivated by the original work [1].

• 2.    Main Results

First and foremost, we shall recall some assertions from [1] and [2] needed for our successful presentation. A group G is said to be p-splitting if G _p is a direct factor of G.

Proposition. Suppose G is a p -splitting group and F is perfect. Then the following hold:

• (i)    If G _p is simply presented, then S(FG)/G _p is simply presented and G _p is a direct factor of S(FG) with simply presented complement. In particular, S(FG) is simply presented if, and only if, G _p is simply presented.

• (ii)    If H is some group such that FG and FH are F-isomorphic, then H _p = G p , provided that G _p is simply presented.

Remark 1. The authors of [14] just have imitated and duplicated the original proofs of (i) from [1] without any new moments but with rather more superfluous explanations which are, in fact, trivial and therefore they were omitted in [1].

Moreover, May, Mollov and Nachev reproved in ([14], Theorem 2 . 7) almost the same statement as that of (ii) but provided G is splitting. However, their proof is manifestly incorrect and obviously has the following serious shortcoming: Using a classical lemme of Ullery, they reduce the general case to the p -mixed case considering the p -mixed groups G/ _q6=p G _q and H/ _q6=p and, besides, they used that G/ _q6=p G _q is splitting. This is, however, not correct since they need to show that if G is splitting then so is G/ _q6=p G q .

This implication is not trivial and follows like this (see, e.g., [7]): Even more, G is p -splitting if, and only if, G/ _q6=p G _q is splitting.

Sketch of proof . Indeed, if G = G p x M , then G = G t M and hence G/ ' q= _p G q = (G t / U q=p G q ) (M(U q=p G q )/ U q=p G q )• But G t / U q=p G q = (G/ U q=p G q ) , and by the modular law we have that

M   Gq q=p

П G t = (Ц G q) [m П G t] = ( Ц G q) [m П (G p x M t)] q=p                    q=p

= Mt( Ц Gq) [m П Gp] = Ц Gq q=p                  q=p since Mt = []q=p Mq C [Jq=p Gq.

Concerning the converse, it has an identical verification as that demonstrated above. So, we are done.

Remark 2. In [14] was claimed also that in reference [4] from [14], i. e. in [Da], is used the fact, which as stated by them is certainly wrong, that if a subgroup B is isomorphic to a direct factor of an abelian group A, then B is a direct factor of A. However, in [1] this was utilized not for an arbitrary subgroup; reciprocally it was used for a special subgroup of V (FG) for which this is really true. No counterexample in [14] is given, only redundant comments that are false.

We next will reprove an assertion of ours first established in [1]. The idea is given in [9] but the proof here is a little more conceptual.

Theorem. Suppose G is a splitting p -mixed abelian group such that G _p is simply pre sented and KH = KG are K-isomorphic for some group H and some field K of prime characteristic p. Then H = G.

Without loss of generality, and as it is well-known, we may assume that KH = KG since KG = KH implies that KG = KH ⁰ for some H = H ⁰ 6 V ( K G ). Moreover, it is well known that K may be chosen to be algebraically closed and hence perfect. Therefore, one may write that A = V ( KG ) = V ( KH ). Suppose L is a perfect field of characteristic p , containing K and having the same identity, such that LA exists. Thus I ( KG ; G ) = I ( KH ; H ) and consequently LA • I (KG; G) = LA • I (KH ; H ), i.e., I (LA; G) = I (LA; H ) (see cf. [10])- Furthermore, L ( A/G ) = LA/I ( LA ; G ) = LA/I ( LA ; H ) = L ( A/H ). But, according to Proposition 2.1 (i), A/G = V ( KG ) /G = S ( KG ) /G _p is simply presented p -torsion. Consequently, by the classical isomorphism result from [13], we deduce that A/H = V ( KH ) /H is simply presented p -torsion as well. Therefore, H is a direct factor of V ( KH ) and hence H _p is a direct factor S ( KH ). But G = G p x M implies that V(KG) = S(KG) x T . Thus V(KH ) = S(KH ) x T and by what we have shown H p is a direct factor of V(KH ), whence of H С V(KH ). Writing G = G p x (G/G _p ) and H = H p x (H/H _p ), it follows that G = H because again with [10] or [11] at hand we derive G _p = H _p and G/G _p = H/H _p . B

Remark 3. Owing to our isomorphism theorem alluded to above, Corollary 4 . 3 and Theorem 4 . 6 both from [14] are unnecessary since they are its elementary consequences.

Moreover, it is not clear to the (non-expert) reader why in the formulation of Theorem 4 . 6, the subgroup G _p must be reduced. The result (perhaps) follows for not necessarily reduced G _p .

And finally, in [14] have criticized that some statements from papers of Danchev (see, e.g., the bibliography in [14]) are not completely proved in the sense that not all details are given in an explicit form. In this way, this is really almost true, but the reason is that these assertions and the missing parts and points are trivial the readers can verify that by seeing Lemmas 2.1–2.5 from [14].