Prove: If H is a finite group of even cardinality, the H contains an element h of ord

apple2009 · #1 $Old$ November 5th, 2009, 07:52 PM

Prove: If H is a finite group of even cardinality, the H contains an element h of order 2.

Drexel28 · #2 $Old$ November 5th, 2009, 08:14 PM

Quote:

Originally Posted by apple2009 $View Post$

Prove: If H is a finite group of even cardinality, the H contains an element h of order 2.

Problem: Suppose $\left|H\right|$ is even. Prove that at least one element of $H$ is of order two.

Proof: Suppose that $H$ contained no elements of order two. We can see then that $a\ne a^{-1}$ for all nontrivial elements of $H$ . Therefore elements of this nature come in distinct pairs ( $a,a^{-1}$ ) adding up all these elements will then give an even number. Then noting that since no nontrivial element of $H$ had order two we can see that the total number of elements in $H$ is the number of all elements whose order isn't two and the identity element. But this is an even number plus one, thus odd which is of course a contradiction.