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Владикавказский математический журнал 2012, Том 14, Выпуск 2, С. 31-34
УДК 512.742
SOME ISOMORPHISM RESULTS ON COMMUTATIVE GROUP ALGEBRAS
P. Danchev
We prove certain results pertaining to some isomorphism properties of commutative modular group
algebras and briefly review a paper by pointing out some obvious mistakes and essential incorrectness.
Key words: group, ring, group algebra, isomorphism, splitting group, p-mixed group, totally projective
group.
1. Introduction
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(FG). Moreover, let S(FG) = Vp(FG) and let Gp 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 Gt is a totally projective (reduced or not) p-group (compare also with the original source [1]). Note that when the torsion subgroup Gt 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 (FG)/G is simply presented and, in particular, G is a direct factor V (FG) with simply presented complement.
© 2012 Danchev P.
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(FG) 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 Gp is a direct factor of G.
Proposition. Suppose G is a p-splitting group and F is perfect. Then the following hold:
(i) If Gp is simply presented, then S (FG)/Gp is simply presented and Gp is a direct factor of S(FG) with simply presented complement. In particular, S(FG) is simply presented if, and only if, Gp is simply presented.
(ii) If H is some group such that FG and FH are F-isomorphic, then Hp = Gp, provided that Gp 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 lemma of Ullery, they reduce the general case to the p-mixed case considering the p-mixed groups G/Uq=p Gq and H/Uq=p and, besides, they used that G/\\q=p Gq is splitting. This is, however, not correct since they need to show that if G is splitting then so is G/ Uq=p Gq.
This implication is not trivial and follows like this (see, e.g., [7]): Even more, G is p-splitting if, and only if, G/ ]Jq=p Gq is splitting.
Sketch of1 proof. Indeed, if G = Gp x M, then G = GtM and hence G/\\q=p Gq =
(G/ u,=p G) (M(U,=p Gq)/U,=p Gq). But Gt/ U,=p Gq = (G/ U,=p Gq), and by the modular law we have that
M( JjGqJ n Gt = m Gq
q=p q=p
= M, H Gq
q=p
M n G,
M Gp
U Gq\ [m n (Gp x Mt)
n-Ln /
q=p
U G’«
q=p
since Mt = Uq=p Mq C Uq=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 [1], 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.
Some Isomorphism Results on Commutative Group Algebras
33
Theorem. Suppose G is a splitting p-mixed abelian group such that Gp is simply presented 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' for some H = H' ^ V(KG). 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)/Gp 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 Hp is a direct factor S(KH). But G = Gp x M implies that V(KG) = S(KG) x T. Thus V(KH) = S(KH) x T and by what we have shown Hp is a direct factor of V(KH), whence of H C V(KH). Writing G = Gp x (G/Gp) and H = Hp x (H/Hp), it follows that G = H because again with [10] or [11] at hand we derive Gp = Hp and G/Gp = H/Hp. >
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 Gp must be reduced. The result (perhaps) follows for not necessarily reduced Gp.
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 absolutely trivial — the readers can verify that by seeing Lemmas 2.1-2.5 from [14].
References
1. Danchev P. V. Isomorphism of modular group algebras of totally projective abelian groups // Commun. Algebra.-2000.-Vol. 28, № 5.—P. 2521-2531.
2. Danchev P. V. Invariants for group algebras of splitting abelian groups with simply presented components // Compt. Rend. Acad. Bulg. Sci.—2002.—Vol. 55, № 2.—P. 5-8.
3. Danchev P. V. Isomorphism of commutative group algebras of closed p-groups and p-local algebraically compact groups // Proc. Amer. Math. Soc.—2002.—Vol. 130, № 7.—P. 1937-1941.
4. Danchev P. V. A note on the isomorphic modular group algebras of abelian groups with simply presented components // Compt. Rend. Acad. Bulg. Sci.—2004.—Vol. 57, № 12.—P. 13-14.
5. Danchev P. V. Isomorphic modular group algebras of semi-complete primary abelian groups // Bull. Korean Math. Soc.—2005.—Vol. 42, № 1.—P. 53-56.
6. Danchev P. V. Invariants of quotient groups in commutative modular group algebras // Proc. Roman. Acad., Ser. A: Math.—2006.—Vol. 7, № 3.—P. 173-176.
7. Danchev P. V. Notes on the isomorphism and splitting problems for commutative modular group algebras // Cubo Math. J.—2007.—Vol. 9, № 1.—P. 39-45.
8. Danchev P. V. Isomorphism of modular group algebras of abelian groups with semi-complete p-primary components // Commun. Korean Math. Soc.—2007.—Vol. 22, № 2.—P. 157-161.
9. Danchev P. V. Notes on the isomorphic modular group algebras of p-splitting and p-mixed abelian groups // Proc. Roman. Acad., Ser. A: Math.—2008.—Vol. 9, № 1.
10. Karpilovsky G. Commutative Group Algebras.—New York: Marcel Dekker, 1983.
11. Karpilovsky G. Unit Groups of Group Rings.—Harlow: Longman Scientific & Technical, 1989.—393 p.
12. Karpilovsky G. Unit groups of commutative group algebras // Expo. Math.—1990.—Vol. 8.—P. 247287.
13. May W. Modular group algebras of simply presented abelian groups // Proc. Amer. Math. Soc.— 1988.—Vol. 104, № 2.—P. 403-409.
14. May W. L., Mollov T. Zh., Nachev N. A. Isomorphism of modular group algebras of p-mixed abelian groups // Commun. Algebra.—2010.—Vol. 38, № 6.—P. 1988-1999.
Received January 12, 2011.
Danchev Peter V.
Plovdiv State University «Paissii Hilendarski», Professor 24 Tzar Assen Street, 4000 Plovdiv, Bulgaria E-mail: [email protected]
ОБ ИЗОМОРФИЗМАХ КОММУТАТИВНЫХ ГРУППОВЫХ АЛГЕБР
Данчев П. В.
Установлены некоторые результаты об изоморфизмах коммутативных групповых алгебр и приводится краткий реферат статьи [14] с указанием очевидных ошибок и существенных неточностей.
Ключевые слова: группа, кольцо, групповая алгебра, изоморфизм, расщепляющая группа, р-сме-шанная группа, проективная группа.
