#### Vol. 7, No. 2, 2014

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On groups with a class-preserving outer automorphism

### Peter A. Brooksbank and Matthew S. Mizuhara

Vol. 7 (2014), No. 2, 171–179
##### Abstract

Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. For each integer $r>1$, there is precisely one 2-group of nilpotency class $r$ in each of the four families. All other known families of 2-groups possessing a class-preserving outer automorphism consist entirely of groups of nilpotency class 2.

##### Keywords
$p$-groups, class-preserving automorphisms, polycyclic groups
##### Mathematical Subject Classification 2010
Primary: 20D15, 20D45, 20E45
##### Milestones
Received: 4 August 2012
Revised: 7 November 2012
Accepted: 17 November 2012
Published: 16 November 2013

Communicated by Nigel Boston
##### Authors
 Peter A. Brooksbank Department of Mathematics Bucknell University 380 Olin Science Building Lewisburg, PA 17837 United States Matthew S. Mizuhara Department of Mathematics Bucknell University 380 Olin Science Building Lewisburg, PA 17837 United States