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

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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