Commuting automorphism of p-groups with cyclic maximal subgroups
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Let G be a group and let p be a prime number. If the setA(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 withcyclic maximal subgroup is an A(G)-group. We also find the structureof the group A(G) and we show that A(G) = Autc(G). Moreover, weprove that for any prime p and all integers n 3, there exists a non-abelian A(G)-group of order pn in which A(G) = Autc(G). If p > 2, then A(G) = Zp × Zpn−2 and if p = 2, then A(G) = Z2 × Z2 × Z2n−3 orZ2 × Z2.
Abstract
Let G be a group and let p be a prime number. If the setA(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 withcyclic maximal subgroup is an A(G)-group. We also find the structureof the group A(G) and we show that A(G) = Autc(G). Moreover, weprove that for any prime p and all integers n 3, there exists a non-abelian A(G)-group of order pn in which A(G) = Autc(G). If p > 2, then A(G) = Zp × Zpn−2 and if p = 2, then A(G) = Z2 × Z2 × Z2n−3 orZ2 × Z2.
참고문헌 (7)
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