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Projective characters of degree one and the inflation-restriction sequence

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

R. J. Higgs
R. J. Higgs
Affiliation:
Department of Mathematics, University College Dublin, Belfield Dublin 4, Ireland
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Abstract

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Let G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.

Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

Keywords

projective representations of finite groups.

MSC classification

Secondary: 20C25: Projective representations and multipliers
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 2 , April 1989 , pp. 272 - 280
Copyright
Copyright © Australian Mathematical Society 1989

References

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