Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer.

Question

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. Prove that, $\exists$ $x\neq e$ such that $x^2=e$ where $e$ represents the identity of $G$.

score 2 · Accepted Answer · answered Jun 25 '15 at 21:53

2

Hint: $$G = \{e\} \cup \{x : x = x^{-1},\,x\ne e\} \cup \{x : x \ne x^{-1} \}$$ and the lattermost set has even cardinality because inversion is an involution on it.

answered Jun 25 '15 at 21:53

Unit

7,601

score 1 · Answer 2 · answered Jun 25 '15 at 21:52

1

Hint: Pair up non-identity elements with their inverse.

answered Jun 25 '15 at 21:52

TomGrubb

12,909

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer.

2 Answers2