# Arbitrary Functions in Group Theory

### From NZJM

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**New Zealand Journal of Mathematics**

**Vol.** 45, (2015), **Pages** 1-9

**Ian Hawthorn**

Department of Mathematics,

University of Waikato,

Hamilton,

New Zealand.

**Yue Guo**

**Abstract** Two measures of how near an arbitrary function between
groups is to being a homomorphism are considered. These have properties
similar to conjugates and commutators. The authors show that there is a
rich theory based on these structures, and that this theory can be used to
unify disparate approaches such as group cohomology and the transfer and to
prove theorems. The proof of the Schur-Zassenhaus theorem is recast in this
context. We also present yet another proof of Cauchy's theorem and
a very quick approach to Sylow's theorem.

**Keywords** Arbitrary functions

**Classification** (MSC2000) 20D99.

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