On groups of odd order admitting an elementary 2-group of automorphisms. (English) Zbl 1242.20030

Oliveira, Karise G.; Shumyatsky, Pavel; Sica, Carmela

Rend. Semin. Mat. Univ. Padova 126, 229-236 (2011).

Let \(G\) be a finite group of odd order with derived length \(k\), acted upon by an elementary Abelian 2-group of automorphisms \(A\) with \(C_G(A)\) of exponent \(e\). The authors, applying the technique developed by P. Shumyatsky and C. Sica, [in Commun. Algebra 38, No. 11, 4188-4192 (2010; Zbl 1216.20017)], establish that the group \(G\) has a normal series \(G=G_0\geq T_0\geq G_1\geq T_1\geq\cdots\geq G_n\geq T_n=1\) such that the quotients \(G_i/T_i\) have \(\{k,e,n\}\)-bounded exponent and the quotients \(T_i/G_{i+1}\) are nilpotent of \(\{k,e,n\}\)-bounded class. Furthermore, if \(A\) is the four-group and the \(c\)-th term of the lower central series of the centralizer of every nontrivial automorphism of \(A\) has exponent \(e\) then it is shown that there is a normal series \(G=T_4\geq T_3\geq T_2\geq T_1\geq T_0=1\) such that the quotients \(T_3/T_2\), \(T_1\) have \(\{e,c,k\}\)-bounded exponent and the quotients \(T_4/T_3\), \(T_2/T_1\) are nilpotent of \(\{e,c,k\}\)-bounded class.

Reviewer: János Kurdics (Nyíregyháza)

MSC:

20D45	Automorphisms of abstract finite groups
20D60	Arithmetic and combinatorial problems involving abstract finite groups
20D30	Series and lattices of subgroups
20D10	Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks

Keywords:

groups of odd order; finite solvable groups; automorphisms; centralizers; exponents; nilpotency classes; normal series; derived lengths

Citations:

Zbl 1216.20017

References:

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