On semicomplete finite p-groups
Tóm tắt
Let G be a finite group and N be a non-trivial proper normal subgroup of G. The pair (G, N) is called a Camina pair if
$$xN\subseteq x^G$$
for all
$$x\in G\setminus N$$
, where
$$x^G$$
denotes the conjugacy class of x in G. Also let
$$\mathrm {Aut}^{G'}(G)$$
denote the group of all automorphisms of G fixing
$$G/G'$$
elementwise. A group G is called semicomplete if
$$\mathrm {Aut}^{G'}(G)=\mathrm {Inn}(G)$$
. In this paper, using the notion of Frattinian groups, we give a necessary and sufficient condition for a finite p-group G such that (G, Z(G)) is a Camina pair to be semicomplete.
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