Question

IfG is a group of even order, prove that contains an element of order 2.

Short answer

We proved that G contains an element of order 2.

Step-by-step solution

To find the element of the group

Let G be a group of even order.

We have to show that G contains an element of order 2.

If possible, assume that G does not contain any element of order 2.

Now, for every $a\in G\backslash \left\{e\right\}$,$\exists$ unique element a^-1 $\in$G such that$a\ne a^{-1}$

and $a{a}^{-1}$$=e$, where e is an identity element.

To prove G  contains an element of order 2

If we consider two distinct elements a_1,,a₂ such that a_1,,a₂$\in$G with ${a_1}^{-1}\ne {{a_2}^{-1}}_{}$,

then .

Therefore, the cardinality of G \$\left\{e\right\}$ is even.

This implies that the order of is G odd, which is a contradiction to our hypothesis.

Hence, G contains an element of order 2.