Every group is an outer automorphism group of a finitely generated group

Let G and F be groups. A G-defining subgroup of F is a normal subgroup N of F such that F/N is isomorphic to G. The automorphism group Aut (F) acts on the set of G-defining subgroups of F. If G is finite and F is finitely generated, one obtains a finite permutation representation of Out (F), the outer automorphism group of F. We study these representations in the case that F is a free group.

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Actions of higher-rank lattices on free groups

Compositio Mathematica ◽

10.1112/s0010437x11005598 ◽

2011 ◽

Vol 147 (5) ◽

pp. 1573-1580 ◽

Cited By ~ 2

Author(s):

Martin R. Bridson ◽

Richard D. Wade

Keyword(s):

Automorphism Group ◽

Free Group ◽

Lie Group ◽

Outer Automorphism ◽

Real Rank ◽

Semisimple Lie Group ◽

Finitely Generated ◽

Outer Automorphism Group ◽

Finite Image ◽

Higher Rank

AbstractIf G is a semisimple Lie group of real rank at least two and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.

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Outer automorphism group of free groups and the outer spaces

Arithmetic Groups and Their Generalizations - AMS/IP Studies in Advanced Mathematics ◽

10.1090/amsip/043/21 ◽

2009 ◽

pp. 179-182

Keyword(s):

Automorphism Group ◽

Free Groups ◽

Outer Automorphism ◽

Outer Automorphism Group

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Continuous trace C*-algebras with given Dixmer-Douady class

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700023661 ◽

1985 ◽

Vol 38 (3) ◽

pp. 394-407 ◽

Cited By ~ 3

Author(s):

Iain Raeburn ◽

Joseph L. Taylor

Keyword(s):

Automorphism Group ◽

Outer Automorphism ◽

Explicit Construction ◽

Irreducible Representations ◽

Outer Automorphism Group ◽

C Algebras ◽

Finite Dimensional ◽

Continuous Trace ◽

Separable Algebras

AbstractWe give an explicit construction of a continuous trace C*algebra with prescribed Dixmier-Douady class, and with only finite-dimensional irreducible representations. These algebras often have non-trivial automorphisms, and we show how a recent description of the outer automorphism group of a stable continuous trace C*algebra follows easily from our main result. Since our motivation came from work on a new notion of central separable algebras, we explore the connections between this purely algebraic subject and C*a1gebras.

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Quasirecognition of E6(q) by the orders of maximal abelian subgroups

Journal of Algebra and Its Applications ◽

10.1142/s0219498818501220 ◽

2018 ◽

Vol 17 (07) ◽

pp. 1850122 ◽

Cited By ~ 1

Author(s):

Zahra Momen ◽

Behrooz Khosravi

Keyword(s):

Finite Group ◽

Automorphism Group ◽

Simple Group ◽

Outer Automorphism ◽

Composition Factor ◽

Outer Automorphism Group ◽

Abelian Subgroups ◽

A Finite Group ◽

Nonabelian Composition Factor

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

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