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Some remarks on a certain class of finite \(p\)-groups. (English) Zbl 0842.20022
In an earlier paper [Math. Scand. 62, No. 2, 153-172 (1988; Zbl 0639.20015)] we constructed for each odd prime \(p\) explicit functions \(f,g : \mathbb{N} \to \mathbb{N}\) (depending on \(p\)) such that whenever \(G\) is a finite \(p\)-group possessing an automorphism with exactly \(p\) fixed points and order \(p^k\), then \(G\) has a normal subgroup of \(\text{index} \leq f(k)\) whose class is \(\leq g(k)\). In the first part of the present paper we do the same for the prime \(p = 2\). As explained in the earlier paper, these results can be seen as a generalisation of the fact due to C. R. Leedham-Green and S. McKay that if \(G\) is a finite \(p\)-group of maximal class then the maximal subgroup \(C_G(\gamma_2(G)/\gamma_4(G))\), where \(\gamma_i(\cdot)\) denotes terms of the lower central series of \(G\), has its class bounded by a number depending only on \(p\).
In the second part of the paper we prove for odd primes \(p\) a theorem concerning certain \(p\)-groups with an automorphism of \(p\)-power order and exactly \(p\) fixed points. As explained in the paper, this theorem can be seen as a direct generalisation of certain numerical criteria, due to N. Blackburn, for ‘non-exceptionality’ of finite \(p\)-groups of maximal class. As the theorem is somewhat technical, we refer to the paper for its precise statement.
Reviewer: I.Kiming (Essen)
MSC:
20D45 Automorphisms of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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