# If $G$ is non-nilpotent and $M$ is non-normal subgroup of $G$, then $|G: M|=p^{\alpha}$?

Let $G$ be a finite and non-nilpotent group. Is there a non-normal maximal subgroup $M$ such that $|G:M|$ is a prime power?

• In fact is there a non-normal maximal subgroup such that $|G: M|$ is prime power. – M.Mazoo Jun 14 '13 at 19:55
• Not necessarily. There is none in $A_6$ for example. – Derek Holt Jun 14 '13 at 20:03

## 1 Answer

No, there need not be a non-normal maximal subgroup of prime power index in a finite group. For instance $A_6$ has no such maximal subgroup as Derek Holt mentioned.

A finite group with quotient isomorphic to the simple group of order 168 is solvable if and only if every maximal subgroup has prime power index. A finite group is supersolvable if and only if every maximal subgroup has prime index. A finite group is nilpotent if and only if every maximal subgroup is normal.

The finite simple groups with even a single maximal subgroup of prime power index are calssified in Guralnick (1983). Generally they are pretty rare, being alternating of degree a prime power, Ln(q) where $(q^n-1)/(q-1)$ is a prime power, or one of 4 sporadic examples (L2(11), M11, M23, or U4(2)).

Two examples of Ln(q): (a) the first has $n=3$, $q=2$, and $(q^n-1)/(q-1)=7$; L3(2) is the simple group of order 168. (b) the second has $n=2$, $q=3$, and $(q^n-1)/(q-1)=8$; L2(7) is also the simple group of order 168. Every maximal subgroup has prime power index (either 7 or 8).