Dihedral trick

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This fact is useful in work leading up to the Classification of finite simple groups

This article describes a method that can be used to prove that two elements inside a group are conjugate

Statement

Statement with symbols

Let x and y be two distinct involutions (elements of order two) in a finite group G. Suppose xy has order m. Then, \langle x, y \rangle is a dihedral group of order 2m, with cyclic subgroup of order m generated by xy and the element x of order two conjugating xy to its inverse.

Related facts

Applications

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 301, Theorem 1.1, Chapter 9 (Groups of even order), Section 1 (Elementary properties of involutions), More info