ISI2018-MMA-15

Question

Let $G$ be a finite group of even order. Then which of the following statements is correct?

• A. The number of elements of order $2$ in $G$ is even
• B. The number of elements of order $2$ in $G$ is odd
• C. $G$ has no subgroup of order $2$
• D.  None of the above.

Answer 1

Answer is B.

Since the group is of even order and the identity is the inverse of itself, therefore, there are odd number of elements (other than identity).

Also, if there is some element which is not inverse of itself, then we can pair such elements with their inverses. This will leave us with odd number of elements which have to be their own inverses.