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On equality of order of a finite \(p\)-group and order of its automorphism group. (English) Zbl 1323.20022

Let \(G\) be a nonabelian finite \(p\)-group. If \(|G|\) divides \(|\operatorname{Aut}(G)|\) then \(G\) is called an LA-group. It is a longstanding and heavily studied conjecture that \(G\) is an LA-group. In this respect it is interesting when \(G\) has few \(p\)-automorphisms; for groups of maximal class this was studied by I. Malinowska [J. Group Theory 4, No. 4, 395-400 (2001; Zbl 0991.20017)].
The paper deals with \(G\) for which \(|G|=|\operatorname{Aut}(G)|\), if \(G\) has cyclic Frattini, then \(G\) is isomorphic to either \(D_8\), \(S_{16}\) or \(M_{2^n}\); furthermore, if \(G\) is of class 2 and has cyclic centre then \(p=2\) and \(\operatorname{Aut}(G)\) has a cyclic subgroup \(C\) together with the group of central automorphisms \(\operatorname{Aut}_c(G)\) generating a subgroup of index 2 in \(\operatorname{Aut}(G)\), and \(\operatorname{Aut}_c(G)\cap C\) is of order 2.

MSC:

20D45 Automorphisms of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Citations:

Zbl 0991.20017

Software:

Cayley
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Full Text: DOI

References:

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