We know if a non-Abelian finite group is simple then G=G', where G' is the subgroup generated by commutators; i.e. for such groups the index of G' in G is 1.
I want to know if one has classified the finite groups G, for which |G:G'|=2.
Saeid Jafari
A class of finite groups G for which |G:G'|=2 is the symmetric groups Sn , n>2; We know for this class of groups, the derived subgroup is the alternating groups An and |Sn:An|=2. But for any prime p, if G belong to the class of non-cyclic p-groups then one can show that |G:G'| is at least p2.
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