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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.

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 |