COMMUTING AUTOMORPHISM OF p-GROUPS WITH CYCLIC MAXIMAL SUBGROUPS Vosooghpour, Fatemeh; Kargarian, Zeinab; Akhavan-Malayeri, Mehri;
Let G be a group and let be a prime number. If the set of all commuting automorphisms of G forms a subgroup of Aut(G), then G is called -group. In this paper we show that any -group with cyclic maximal subgroup is an -group. We also find the structure of the group and we show that . Moreover, we prove that for any prime and all integers , there exists a non-abelian -group of order in which . If > 2, then and if , then or .
commuting automorphism;cyclic maximal subgroup;
M. Deaconescu, Gh. Silberberg, and G. L. Walls, On commuting automorphisms of groups, Arch. Math. (Basel) 79 (2002), no. 6, 423-429.
M. Deaconescu and G. L. Walls, Right 2-Engel elements and commuting automorphism of group, J. Algebra 238 (2001), no. 2, 479-484.
D. S. Dummit and R. M. Foote, Abstract Algebra, Prentice-Hall, Inc, 1991.
I. N. Herstein, T. J. Laffey, Problems and solutions: Solutions of elementary problems: E3039, Amer. Math. Monthly 93 (1986), no. 10, 816-817.
Z. Kargarian and M. Akhavan Malayeri, On the commuting automrphisms of groups of order $p^3$, Adv. Appl. Math. Sci. 9 (2011), no. 2, 115-120.
D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
F. Vosooghpour and M. Akhavan-Malayeri, On commuting automorphisms of p-groups, Comm. Algebra 41 (2013), no. 4, 1292-1299.