Representation of group isomorphisms I

Group Homomorphisms and Isomorphisms

c W W L Chen, 1991, 1993, 2013. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible t...

Natural Isomorphisms in Group Theory.

Note on the Group of Isomorphisms*

LET SV a2, ••', sg represent all the operators of a group G and let tasa correspond to sa (a = 1, 2, •••, g) in any given simple isomorphism of G with itself. I t is evident that ta is some operator of G. When G is abelian these £a's must constitute a group T which is isomorphic with (?.* In this isomorphism, ta evidently can not be the inverse of sa unless sa = 1. As this condition is sufficie...

Isomorphisms of p-adic group rings

A long-standing problem, first posed by Graham Higman [15] and later by Brauer [4] is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG = ZH implies G 2: H? Many authors have worked on this question, but progress has been difficult [30]. Perhaps the best positive result was that of Whitcomb in 1968 [37], who showed that the implication G = H hol...

Cocycle Deformations and Brauer Group Isomorphisms

Let H be a Hopf algebra over a commutative ring k with unity and σ : H ⊗ H −→ k be a cocycle on H. In this paper, we show that the Yetter-Drinfeld module category of the cocycle deformation Hopf algebra H is equivalent to the Yetter-Drinfeld module category of H. As a result of the equivalence, the “quantum Brauer” group BQ(k,H) is isomorphic to BQ(k,H). Moreover, the group Gal(HR) constructed ...