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

[John Dixon]

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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