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The Scientific World Journal
Volume 2014, Article ID 850526, 3 pages
http://dx.doi.org/10.1155/2014/850526
Research Article

The Structure of EAP-Groups and Self-Autopermutable Subgroups

1Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad 9187147578, Iran
2Department of Mathematics, Khayyam University, Mashhad 9189747178, Iran
3Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Iran

Received 19 May 2014; Accepted 16 September 2014; Published 11 December 2014

Academic Editor: K. C. Sivakumar

Copyright © 2014 Shima Housieni and Mohammad Reza Rajabzadeh Moghaddam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A subgroup H of a given group G is said to be autopermutable, if for all . We also call H a self-autopermutable subgroup of G, when implies that . Moreover, G is said to be EAP-group, if every subgroup of G is autopermutable. One notes that if α runs over the inner automorphisms of the group, one obtains the notions of conjugate-permutability, self-conjugate-permutability, and ECP-groups, which were studied by Foguel in 1997, Li and Meng in 2007, and Xu and Zhang in 2005, respectively. In the present paper, we determine the structure of a finite EAP-group when its centre is of index 4 in G. We also show that self-autopermutability and characteristic properties are equivalent for nilpotent groups.