# a group of even order contains an element of order 2

###### Proposition.

Every group of even order contains an element of order $\mathrm{2}$.

###### Proof.

Let $G$ be a group of even order, and consider the set $S=\{g\in G:g\ne {g}^{-1}\}$. We claim that $|S|$ is even; to see this, let $a\in S$, so that $a\ne {a}^{-1}$; since ${({a}^{-1})}^{-1}=a\ne {a}^{-1}$, we see that ${a}^{-1}\in S$ as well. Thus the elements of $S$ may be exhausted by repeatedly selecting an element and it with its inverse^{}, from which it follows that $|S|$ is a multiple (http://planetmath.org/Divisibility) of $2$ (i.e., is even). Now, because $S\cap (G\setminus S)=\mathrm{\varnothing}$ and $S\cup (G\setminus S)=G$, it must be that $|S|+|G\setminus S|=|G|$, which, because $|G|$ is even, implies that $|G\setminus S|$ is also even. The identity element^{} $e$ of $G$ is in $G\setminus S$, being its own inverse, so the set $G\setminus S$ is nonempty, and consequently must contain at least two distinct elements; that is, there must exist some $b\ne e\in G\setminus S$, and because $b\notin S$, we have $b={b}^{-1}$, hence ${b}^{2}=1$. Thus $b$ is an element of order $2$ in $G$.
∎

Notice that the above is logically equivalent to the assertion that a group of even order has a non-identity element that is its own inverse.

Title | a group of even order contains an element of order 2 |
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Canonical name | AGroupOfEvenOrderContainsAnElementOfOrder2 |

Date of creation | 2013-03-22 17:11:37 |

Last modified on | 2013-03-22 17:11:37 |

Owner | azdbacks4234 (14155) |

Last modified by | azdbacks4234 (14155) |

Numerical id | 20 |

Author | azdbacks4234 (14155) |

Entry type | Theorem |

Classification | msc 20A05 |