Abstract. Let G be a group and let p be a prime number. If the set A(G) of all commuting automorphisms of G forms a subgroup of Aut(G), then G is called A(G)-group. In this paper we show that any p-group with cyclic maximal subgroup is an A(G)-group. We also find the structure of the group A(G) and we show that A(G) = Autc(G). Moreover, we prove that for any prime p and all integers n ≥ 3, there exists a nonabelian A(G)-group of order p n in which A(G) = Autc(G). If p > 2, then A(G) ∼ = Zp × Z p n−2 and if p = 2, then A(G) ∼ = Z 2 × Z 2 × Z 2 n−3 or Z 2 × Z 2 .
Let [Formula: see text] be a group. If the set [Formula: see text] for all [Formula: see text] forms a subgroup of [Formula: see text], then [Formula: see text] is called [Formula: see text]-group. Let [Formula: see text] be an odd prime. Recently it has been proven that a finite [Formula: see text]-group of almost maximal class is an [Formula: see text]-group. For finite 2-groups of almost maximal class, the situation is much more complicated. This paper deals with the case [Formula: see text]. We prove that a 2-group of almost maximal class is an [Formula: see text]-group. We show the minimum coclass of a non-[Formula: see text], [Formula: see text]-group is equal to 3. We also discuss the direct product of certain [Formula: see text]-groups and subsequently we give some applications of our results.
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