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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