Involution

:See involution (philosophy) for the philosophy meaning.

Involutions in group theory

In group theory, an element of a group is an involution if it has order 2; i.e. an involution is an element a such that a2 = e, where e is the identity element. Originally, this definition differed not at all from the first definition above, since members of groups were always bijections from a set into itself, i.e., group was taken to mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution. The group of bijections generated by an involution through composition, is isomorphic with cyclic group C2.

Related Topics:
Group theory - Group - Order - Identity element - Permutation group - Cyclic group

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A permutation is an involution precisely if it can be written as a product of non-overlapping transpositions.

Related Topics:
Permutation - Transposition

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The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups.

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Coxeter groups are groups generated by their involutions. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.

Related Topics:
Coxeter group - Regular polyhedra - Generalizations to higher dimensions

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