Question

If $\mathbf{\left|}\mathbf{G}\mathbf{\right|}$is even, prove that G contains an element of order 2.

Short answer

It is proved that, G contains an element of order 2.

Step-by-step solution

Referring to result of Exercise 28

If $a\in G$then $\left|a\right|=\left|a^{-1}\right|$.

Given that localid="1654262803181" $\left|G\right|$is even.

Proving that G contains an element of order 2

We know any element and its inverse have the same order.

Suppose, if we pair each element of G with an order larger than two with its inverse, there must be an even number of elements of G with an order greater than two.

As we know $\left|G\right|$ is even, therefore G has an odd number of non-identity elements.

This implies, G must have an element with order 2.

Hence, G contains an element of order 2.