Dihedral trick

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

• Involutions are either conjugate or have an involution centralizing both of them

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}