'Prove that if a finite group G has only one maximal subgroup M, then |G| is the power of a prime' I've somehow deduced that no finite group has only one maximal subgroup, and I'm having trouble seeing where I went wrong. This is what I have: Let [tex]H_1[/tex] be a subgroup of G. Either [tex]H_1[/tex] is maximal and equal to M or it is not maximal and there is a [tex]H_2[/tex] such that [tex]H_1<H_2<G[/tex] (using < to mean proper subgroup). Apply the same argument to [tex]H_2[/tex] and we get an ascending chain of subgroups. Since G is finite the process must end eventually (when we reach M). Thus every subgroup of G is a subgroup of M and hence every element of G is an element of M, and G=M, a contradiction. So would someone like to point out the flaw in the above reasoning? thanks.