Inverse Conjugacies and Reversing Symmetry Groups

by Geoffrey R. Goodson

American Mathematical Monthly
January, 1999

Subject classification(s): Algebra and Number Theory | Abstract Algebra | Groups | Differential & Difference Equations | Dynamical Systems
Applicable Course(s): 4.2 Mod Algebra I & II

Let $$G$$ be a group and $$C(a)$$ be the centralizer of $$a\in G$$.  The author studies the properties of the skew centralizer $$B(a)=\{x\in G : xa=a^{-1}x \}$$ and the reversing symmetry group $$E(a)=B(a)\cup C(a)$$ of $$a$$.  Many properties provide nice exercises for an introductive course of abstract algebra.  The author also shows the dynamical origin and applications of such algebraic structures, which arise naturally from the ergodic theory of measure-preserving transformations.

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