group theory question

**tukilala** · June 9th 2009, 10:15 AM

prove that if G is a finite group that the number of elements in G is even
So
e≠x belong to G (e is the neutral element)
that provide x^2 = e (in other words: x=x^-1).

**flyingsquirrel** · June 9th 2009, 10:47 AM

Hello,

Let $A=\left\{x\ :\ x\in G\ \text{and}\ x\neq x^{-1}\right\}$ and $B=\left\{x\ :\ x\in G\ \text{and}\ x=x^{-1}\right\}$ . We have $A\cap B=\emptyset$ and $G=A\cup B$ so $|G|=|A|+|B|$ . Can you show that $|B|$ is even and that there exists an element $x\neq e$ lying in $B$ ?

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