# Maximal subgroup and representations (principal part)

Let $G$ be a finite group, $V$ an irreducible complex representation, $H$ a subgroup.
Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$.

Question: Is $dim(V^H) \le 1$ if $H$ is a maximal subgroup of $G$ ?

Remark : If $H = \{ e \}$ then $G = \mathbb{Z}_p$ and $dim(V)=1$.

#### Solutions Collecting From Web of "Maximal subgroup and representations (principal part)"

$\dim(V^H) = [ \chi_H, 1_H ] = [ \chi, 1_H^G ]$ and it is a bit rare for a primitive permutation character to have a repeated factor, but not too uncommon.

There are several examples amongst simple groups: $G=L_2(11)$, $L_2(13)$, $L_2(17)$, $L_2(19)$, $L_3(3)$, $L_2(23)$, $L_2(25)$, $M_{11}$, $L_2(27)$, $L_2(29)$, $L_2(31)$, $Sz(8)$, $M_{12}$, $J_1$, $A_9$, $L_3(5)$, $J_2$, $L_2(109)$, $L_2(113)$, …, $A_{10}$, $A_{12}$, $A_{13}$, ${}^2F_4(2)’$ for example.

For $G=L_2(11)$ we can take $H$ to be a Sylow 3-normalizer, and then one of the irreducible representations $W$ of dimension 10 (there are two, but only one works; the one with trace -1 on an element of order 6) works. This one might be a bit easier to check. In this example, one can even find $g$ so that $H \cap H^g = 1$ providing a counterexample to the “dual question.”

For $G=A_{13}$ we can take $H$ to be a Sylow 13-normalizer, and then almost every irreducible representation of $G$ works (all except the ones of degree 1, 12, 65, and 66).

Here is the list of counter-examples with $G$ simple and $|G|<30000$ (to be read like $[|G:H|, G,H]$):

[ 55, PSL(2,11), Group([ (1,10)(2,9)(3,4)(5,6)(7,11)(8,12), (1,6,8,2,3,11)(4,9,12,5,10,7) ]) ],
[ 78, PSL(2,13), Group([ (1,14)(4,12)(5,6)(7,10)(8,13)(9,11), (2,8)(3,12)(4,5)(6,10)(9,14)(11,13) ]) ],
[ 91, PSL(2,13), Group([ (1,7)(3,12)(4,14)(5,9)(6,13)(10,11), (1,9)(2,10)(3,12)(4,7)(6,11)(8,13) ]) ],
[ 91, PSL(2,13), Group([ (1,3,8)(2,7,12)(4,5,14)(10,11,13), (1,7)(2,8)(4,9)(5,14)(6,11)(10,13) ]) ],
[ 136, PSL(2,17), Group([ (1,3,13,2,17,4,8,12,15)(5,14,10,9,7,6,18,11,16),(1,4)(2,13)(3,17)(6,7)(8,15)(9,18)(10,11)(14,16) ]) ],
[ 153, PSL(2,17), Group([ (1,10)(2,18)(3,12)(4,17)(6,14)(8,11)(9,16)(13,15), (2,11,3,12,8,18,15,13)(4,6,7,14,17,16,5,9) ]) ],
[ 171, PSL(2,19), Group([ (1,19)(2,11)(3,20)(4,12)(5,17)(6,15)(7,13)(8,18)(9,10)(14,16), (1,15)(2,6)(3,4)(5,11)(7,20)(8,12)(9,13)(10,16)(14,19)(17,18) ]) ],
[ 190, PSL(2,19), Group([ (1,4)(2,7)(3,15)(5,9)(6,20)(8,14)(10,17)(11,13)(12,19)(16,18), (1,8)(2,7)(3,4)(5,15)(6,13)(9,20)(10,18)(11,12)(14,16)(17,19) ]) ],
[ 234, PSL(3,3), Group([ (1,5,3)(2,6,9)(4,7,13)(8,11,10), (1,3,12,5)(2,8,10,6)(4,7)(9,11) ]) ],
[ 253, PSL(2,23), Group([ (1,7,9)(2,21,23)(3,19,16)(4,10,20)(5,14,11)(6,18,17)(8,15,22)(12,24,13), (1,4,23,18)(2,20,15,12)(3,17,21,13)(5,7,6,16)(8,14,19,24)(9,11,22,10) ]) ],
[ 253, PSL(2,23), Group([ (1,10,24)(2,11,3)(4,15,13)(5,23,6)(7,12,8)(9,22,20)(14,17,19)(16,18,21), (1,14,12,2)(3,6,22,10)(4,20,23,16)(5,11,7,18)(8,19,15,21)(9,13,17,24) ]) ],
[ 253, PSL(2,23), Group([ (1,16)(2,17)(3,9)(4,14)(5,10)(6,23)(7,19)(8,24)(11,20)(12,15)(13,22)(18,21), (1,9)(2,19)(3,14)(4,6)(5,13)(7,16)(8,18)(10,20)(11,17)(12,21)(15,23)(22,24) ]) ],
[ 276, PSL(2,23), Group([ (1,6)(2,16)(3,23)(4,13)(5,15)(7,17)(8,14)(9,18)(10,24)(11,22)(12,20)(19,21), (1,9)(2,24)(3,4)(5,11)(6,22)(7,8)(10,15)(12,20)(13,21)(14,16)(17,19)(18,23) ]) ],
[ 300, PSL(2,25), Group([ (1,14,22,19,7,2,9,8,3,11,15,13,12)(4,5,26,21,16,25,10,6,23,17,24,18,20), (2,3)(4,5)(7,11)(8,9)(10,23)(12,14)(13,22)(15,19)(16,24)(17,25)(18,21)(20,26) ]) ],
[ 325, PSL(2,25), Group([ (1,17)(2,10)(3,16)(4,21)(5,12)(6,23)(7,14)(8,22)(11,15)(13,20)(19,25)(24,26), (1,7)(2,18)(3,17)(4,5)(6,23)(8,19)(9,14)(10,11)(12,20)(13,26)(15,16)(22,24) ]) ],
[ 165, M11, Group([ (1,8,5,4,9,7,3,11)(2,6), (1,7)(3,5)(6,10)(8,9) ]) ],
[ 351, PSL(2,27), Group([ (1,14)(2,10)(3,13)(4,18)(5,7)(6,21)(8,16)(9,22)(11,19)(12,24)(15,26)(17,27)(20,23)(25,28), (1,26)(2,6)(3,19)(4,23)(5,14)(7,24)(8,25)(9,16)(10,12)(11,15)(13,28)(17,21)(18,27)(20,22) ]) ],
[ 378, PSL(2,27), Group([ (1,24,25,16,11,4,5,9,26,6,8,18,21)(2,7,10,20,22,27,19,23,12,17,13,28,3), (1,23)(2,9)(3,26)(4,10)(5,7)(6,28)(8,13)(11,20)(12,21)(14,15)(16,22)(17,18)(19,24)(25,27) ]) ],
[ 819, PSL(2,27), Group([ (1,28)(2,10)(3,23)(4,6)(5,25)(7,21)(8,13)(9,15)(11,14)(12,20)(16,19)(17,26)(18,22)(24,27), (1,8,2)(3,27,5)(4,25,19)(6,7,23)(9,10,17)(11,14,12)(13,22,26)(15,18,28)(16,24,21) ]) ]
[ 406, PSL(2,29), Group([ (1,7)(3,22)(4,9)(5,16)(6,21)(10,26)(11,12)(13,19)(14,17)(15,23)(18,25)(20,30)(24,28)(27,29), (1,22)(2,16)(3,24)(4,21)(5,19)(6,17)(7,14)(8,27)(9,29)(11,30)(12,18)(13,23)(15,25)(20,26) ]) ],
[ 435, PSL(2,29), Group([ (1,11)(2,28)(3,14)(4,23)(5,27)(6,12)(7,16)(8,25)(9,22)(10,18)(15,20)(17,30)(19,24)(21,29), (1,15)(2,28)(3,4)(5,22)(7,27)(8,16)(10,21)(11,12)(13,29)(14,30)(17,19)(18,23)(20,25)(24,26) ]) ],
[ 465, PSL(2,31), Group([ (1,24)(2,17)(3,4)(5,29)(6,12)(7,8)(9,19)(10,28)(11,18)(13,31)(14,16)(15,20)(21,26)(22,32)(23,30)(25,27), (1,4)(2,6)(3,15)(5,20)(7,25)(8,22)(9,31)(10,11)(12,23)(13,26)(14,27)(16,21)(17,28)(18,32)(19,24)(29,30) ]) ],
[ 496, PSL(2,31), Group([ (1,27)(2,17)(3,29)(4,10)(5,14)(6,24)(7,22)(8,20)(9,30)(11,15)(12,28)(13,21)(16,32)(18,19)(23,31)(25,26), (1,26)(2,25)(3,23)(4,7)(5,22)(6,31)(8,24)(9,30)(10,32)(11,19)(12,21)(13,27)(14,17)(15,20)(16,29)(18,28) ]) ],
[ 620, PSL(2,31), Group([ (1,26,25,10)(2,30,15,14)(3,9,12,13)(4,18,22,6)(5,32,20,8)(7,27,28,31)(11,23,19,24)(16,21,17,29), (1,2)(3,4)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) ]) ],
[ 620, PSL(2,31), Group([ (1,21,10)(2,4,23)(5,12,15)(6,24,26)(7,19,25)(8,18,20)(9,29,32)(13,17,28)(14,30,22)(16,27,31), (1,17)(2,22)(3,8)(4,21)(5,18)(6,31)(7,13)(9,10)(11,15)(12,20)(14,16)(19,27)(23,25)(24,28)(26,32)(29,30) ]) ],
[ 666, PSL(2,37), Group([ (1,21)(2,9)(3,14)(4,23)(5,31)(6,22)(7,26)(8,24)(11,34)(12,18)(13,28)(15,33)(16,27)(17,38)(19,32)(25,35)(29,30)(36,37), (1,32)(3,35)(4,20)(5,27)(6,13)(7,15)(8,34)(9,12)(10,25)(11,16)(14,31)(17,19)(18,29)(21,22)(23,36)(24,30)(26,38)(28,37) ]) ],
[ 703, PSL(2,37),Group([ (1,6,9,33,12,13,32,15,16,31,19,22,2,21,17,14,11,7)(3,37,28,35,24,4,29,36,27,25,34,8,10,5,23,18,20,30), (1,32)(3,35)(4,20)(5,27)(6,13)(7,15)(8,34)(9,12)(10,25)(11,16)(14,31)(17,19)(18,29)(21,22)(23,36)(24,30)(26,38)(28,37) ]) ],
[ 2109, PSL(2,37), Group([ (1,3)(2,31)(4,9)(5,34)(6,27)(7,28)(8,19)(10,22)(11,14)(12,24)(13,21)(15,26)(16,32)(17,35)(18,38)(20,23)(25,30)(29,36), (1,12,13)(2,37,38)(3,36,5)(6,35,23)(7,22,28)(8,27,15)(9,14,11)(16,26,20)(17,19,32)(18,31,33)(21,25,29)(24,34,30) ]) ],
[ 560, Sz(8), <permutation group of size 52 with 2 generators> ],
[ 1456, Sz(8), <permutation group of size 20 with 2 generators> ],
[ 2080, Sz(8), <permutation group of size 14 with 2 generators> ]