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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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