as part of a referate i prepare, i have to do an induction proof about the group order of non solvable groups.

however the publication i have to present proceeds this proof without giving a starting case and neither did i easily find one in my litarature.

maybe this is a very trivial question, but im not that familiar with this and for me it isnt.

so could anyone help me finding a start for my induction? maybe a dyhedral group?

in addition i need to show that for this Group that G = G'(where G' is the Commutatorgroup) and also ({y}, G) = G for any y from G(in my paper this is called commutator simple, but i got no clue, if this is a common definition), the group should also not be too huge ;)

any help would be appreciated.