# [GAP Forum] Counting number of involutions in finite simple groups.

John Dixon jdixon at math.carleton.ca
Thu Apr 20 17:54:45 BST 2006

```The condition you ask for will never happen.  If G is a finite group of
order g and p^k is a prime power dividing g then a modified version of
the Sylow theorems shows that the number of subgroups of G of order p^k
is congruent to 1 mod p.  In your case, the number of subgroups of order
2 is odd, and this implies that the number of involutions (including 1)
is even.
- John Dixon.

muniru asiru wrote:
>
> I will be grateful if  you can offer some help on this
> problem.
>
> I have been counting the number of involutions in
> finite simple groups and
> I like to know whether or not it is possible to use
> Gap to find examples of finite simple groups for which
>
>
> "the number of involutions including the identity
> element is a prime number (greater or equal to 5)"
>
> or to relax the condition a little bit, can one use
> Gap to find examples of finite simple groups for which
>
> "the number of involutions including the identity
> element is odd (greater or equal to 5)".
>
> If G is a group, x in G is called an involution if
> x^2=1, where 1 is the identity element in G.
>
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```