23

For which finite groups $M$ is it the case that every finite group $G$ with $M$ as a maximal subgroup solvable?

If $M$ satisfies this condition then $M$ is solvable. Also, if $M$ is abelian then $M$ satisfies this condition. Futhermore, I believe that if $M$ is nilpotent and if all 2-subgroups of $M$ are normal subgroups of $M$ (if Sylow 2-subgroups of $M$ are abelian or quaternion, for example) then $M$ satisfies this condition (proof below).

More specific questions:

1) Is there a non-nilpotent group that satisfies this condition?

2) Which 2-groups satisfy this condition?

Apparently, the dihedral group of order 8 satisfies this condition (see Mikko Korhonen's comment on this post). Also, if $M\times N$ satisfies this condition then $M$ and $N$ both satisfy this condition.

(This proof is adapted from j.p.'s answer to the linked question). Let $G$ be minimal such that $G$ is not solvable and such that $G$ contains a maximal subgroup $M$ that is nilpotent and whose 2-subgroups are normal. If $M$ contains a nontrivial normal subgroup $N$ of $G$ then $G/N$ contradicts the minimality of $G$. Thus, $M$ does not contain nontrivial normal subgroups of $G$. In particular, $N_G(P)=M$ for all Sylow $p$-subgroups $P$ of $M$. Then $P$ is a Sylow $p$-subgroup of $N_G(P)$ so $P$ is a Sylow $p$-subgroup of $G$. This shows that $M$ is a Hall subgroup of $G$.

If $P$ is a Sylow $p$-subgroup of $M$ and if $Q$ is a nontrivial normal subgroup of $P$ then $N_G(Q)=M$ which has a normal $p$-complement. For $p=2$, Frobenius' normal $p$-complement theorem gives that $G$ has a normal $p$-complement. For $p\geq3$, Thompson's normal $p$-complement theorem or Glauberman's normal $p$-complement theorem gives that $G$ has a normal $p$-complement (since you only have to consider characteristic $p$-subgroups).

Thus, for each prime $p$ dividing the order of $M$, $G$ has a normal $p$-complement. Then $M$ has a normal complement $N$ in $G$. Since $M$ is solvable but $G$ is not solvable, $N$ is not solvable. In particular, $N$ does not admit a fixed-point-free automorphism of prime order. If $m\in Z(M)$ has prime order then $C_N(m)$ is nontrivial. Then $C_N(m)M$ is a subgroup of $G$ that properly contains $M$ so $C_N(m)M=G$ by the maximality of $M$. Comparing cardinalities shows that $C_N(m)=N$ so $m\in Z(G)$. Then $\langle m\rangle$ is a nontrivial normal subgroup of $G$ contained in $M$ which is a contradiction.

Thomas Browning
  • 4,541
  • A famous example here is when $M$ is a $p$-group for an odd prime $p$. – Steve D Sep 21 '18 at 06:41
  • I wonder if there is a reduction to the case where $G$ is almost simple? That is, could we show that $M$ "forces solvability as a maximal subgroup" if and only if $M$ is solvable and $M$ is not a maximal subgroup of any almost simple group? If the answer is yes, then your questions are answered by a paper of Li and Zhang from 2011 (Proceedings of the LMS), who classify the solvable maximal subgroups of almost simple groups. – Mikko Korhonen Sep 27 '18 at 01:50
  • 2
    About 2), just to give an example: if $p = 2^n - 1$ is a Mersenne prime ($n > 3$), then $PSL(2,p)$ has the dihedral group of order $2^{n-1}$ as a maximal subgroup. – Mikko Korhonen Sep 27 '18 at 01:52
  • Interesting. I computed maximal subgroups of a number of simple groups a while ago. It looked like all small nontrivial semi-direct products of two cyclic groups occurred with the exception of the dihedral group of order 8. I still don't know whether the dihedral group of order 8 forces solvability. – Thomas Browning Sep 27 '18 at 02:21
  • @ThomasBrowning: For your question 1), perhaps the non-trivial semidirect product $C_3 \rtimes C_4$ is an example. – Mikko Korhonen Sep 27 '18 at 03:25
  • 1
    @ThomasBrowning: In "A condition for the solvability of a finite group" (1961) Deskins shows that if $G$ has a nilpotent maximal subgroup of nilpotency class $\leq 2$, then $G$ is solvable. In particular, the dihedral group of order $8$ forces solvability as a maximal subgroup – Mikko Korhonen Sep 27 '18 at 03:46
  • With $M_G:=\cap_{g\in G} M^g$, corollary 4.5 in On locally finite groups with locally nilpotent maximal subgroups by B. Bruno and S Schuur (Arch. Math. Vol 61, pp. 215-20) shows that $M$ nilpotent and $M_G$ not a $2$-group implies the solvability of $G$. Without conditions on $M_G$ one can take direct products with the simple group $PSL_2(17)$ (whose $2$-Sylow subgroups isomorphic to $D_{16}$ are maximal subgroups) with other groups for non-solvable constructions. – j.p. Oct 17 '18 at 05:57
  • Can the condition on $M_G$ be strengthened to get a condition on $M$ that does not depend on $G$? – Thomas Browning Oct 17 '18 at 15:27
  • No. From the condition $M$ maximal in $G$ you get only an handle on everything between $M_G$ and $G$, so from a group-theoretic point one should always assume $M_G=1$. You cannot control direct factors: You cannot get rid of $H$ in $M\times H$ maximal in $G\times H$ without factoring out $M_G$. OK, you could require $M$ not to be the direct product of two non-trivial subgroups, but this cannot be used in this situation. On the contrary with the assumption $M_G=1$, $M$ being direct product (e.g. being product of its $p$-Sylow subgroups for different $p$ if it is nilpotent) is quire useful. – j.p. Oct 17 '18 at 19:26

0 Answers0