The general FF-module theorem. (English) Zbl 1262.20003
Let \(G\) be a group with \(O_p(G)=F^*(G)\) and \(S\) be a Sylow \(p\)-subgroup. If \([J(S),\Omega_1(Z(O_p(G)))]=1\), then \(G=C_G(\Omega_1(Z(S)))N_G(J(S))\). If \([J(S),\Omega_1(Z(O_p(G)))]\neq 1\), then there is a maximal elementary Abelian subgroup \(A\) of \(S\) with \([A,\Omega_1(Z(O_p(G)))]\neq 1\) and
\[
|A/C_A(\Omega_1(Z(O_p(G))))|\geq |\Omega_1(Z(O_p(G)))/C_{\Omega_1(Z(\Omega_p(G)))}(A)|.
\]
In fact by the well-known Thompson-replacement one also may choose \(A\) with \(|\Omega_1 (Z(O_p(G))),A,A]=1\). Hence we call an \(\mathbb{F}_p\)-module \(V\) for a group \(M\) an FF-module (failure of factorization) if there is an elementary Abelian \(p\)-subgroup \(A\) of \(G\) such that \([A,V]\neq 1\) and \(|V/C_V(A)|\leq|A/C_A(V)|\).
There is a long history of papers dealing with irreduble FF-modules. Using the relation to quadratic modules J. G. Thompson [Actes Congr. internat. Math. 1970, 1, 375-376 (1971; Zbl 0236.20024)] and A. A. Premet and I. D. Suprunenko [Math. Nachr. 110, 65-96 (1983; Zbl 0522.20027)] determined these modules for \(p\geq 5\). In [Commun. Algebra 19, No. 12, 3193–3222 (1991; Zbl 0822.20010)] Th. Meixner determined the FF-modules for groups of Lie type in characteristic \(3\) and \(p=3\). For \(p=2\) and \(M\) a group of Lie type in characteristic \(2\) this has been done by B. N. Cooperstein [Commun. Algebra 6, 1239-1288 (1978; Zbl 0377.20039)]. Lateron Guralnick, Lawther and Malle received as a corollary of a classification of the 2F-modules for nearly simple groups also one for FF-modules [R. M. Guralnick and G. Malle, J. Algebra 257, No. 2, 348-372 (2002; Zbl 1017.20005); R. M. Guralnick, R. Lawther and G. Malle, J. Algebra 307, No. 2, 643-676 (2007; Zbl 1115.20002); R. M. Guralnick and G. Malle, Finite groups 2003. Proceedings of the Gainesville conference on finite groups, Gainesville, USA, 2003. Berlin: Walter de Gruyter. 117-183 (2004; Zbl 1085.20004)]. Recall that the paper of J. G. Thompson [loc. cit.] and so the classification of the irreducible FF-modules for \(p\geq 5\) is independent of the classification of the finite simple groups.
In this paper, which depends on the classification of the finite simple groups, the authors drop the assumption of irreducibility and focus more on the possible subgroups \(A\) and the action on \(V\). The results are technical and long, so they cannot be stated here.
There is a long history of papers dealing with irreduble FF-modules. Using the relation to quadratic modules J. G. Thompson [Actes Congr. internat. Math. 1970, 1, 375-376 (1971; Zbl 0236.20024)] and A. A. Premet and I. D. Suprunenko [Math. Nachr. 110, 65-96 (1983; Zbl 0522.20027)] determined these modules for \(p\geq 5\). In [Commun. Algebra 19, No. 12, 3193–3222 (1991; Zbl 0822.20010)] Th. Meixner determined the FF-modules for groups of Lie type in characteristic \(3\) and \(p=3\). For \(p=2\) and \(M\) a group of Lie type in characteristic \(2\) this has been done by B. N. Cooperstein [Commun. Algebra 6, 1239-1288 (1978; Zbl 0377.20039)]. Lateron Guralnick, Lawther and Malle received as a corollary of a classification of the 2F-modules for nearly simple groups also one for FF-modules [R. M. Guralnick and G. Malle, J. Algebra 257, No. 2, 348-372 (2002; Zbl 1017.20005); R. M. Guralnick, R. Lawther and G. Malle, J. Algebra 307, No. 2, 643-676 (2007; Zbl 1115.20002); R. M. Guralnick and G. Malle, Finite groups 2003. Proceedings of the Gainesville conference on finite groups, Gainesville, USA, 2003. Berlin: Walter de Gruyter. 117-183 (2004; Zbl 1085.20004)]. Recall that the paper of J. G. Thompson [loc. cit.] and so the classification of the irreducible FF-modules for \(p\geq 5\) is independent of the classification of the finite simple groups.
In this paper, which depends on the classification of the finite simple groups, the authors drop the assumption of irreducibility and focus more on the possible subgroups \(A\) and the action on \(V\). The results are technical and long, so they cannot be stated here.
Reviewer: Gernot Stroth (Halle)
MSC:
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
Keywords:
finite groups; representations; groups of local characteristic \(p\); irreducible failure of factorization modulesCitations:
Zbl 0236.20024; Zbl 0522.20027; Zbl 0822.20010; Zbl 0377.20039; Zbl 1017.20005; Zbl 1115.20002; Zbl 1085.20004References:
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