# A criterion for maximality in $SU_n$.

Is every primitive self-normalizing subgroup of $$SU_n$$ maximal?

Here by maximal I mean maximal among proper closed subgroups.

This is true for $$SU_2$$. The only self-normalizing subgroups of $$SU_2$$ are the binary octahedral group, the binary icosahedral group, and $$O_2(\mathbb{R})$$. Of these, only the binary octahedral and binary icosahedral group are primitive.

Recall that we say a subgroup $$G$$ of $$GL_n(\mathbb{C})$$ is imprimitive if we can write $$\mathbb{C}^n=V_1 \oplus \dots \oplus V_k$$ as a direct sum of smaller subspaces such that every $$g \in G$$ just permutes the subspaces. In other words, for any $$g \in G$$ the subspaces $$g(V_1) \oplus \dots \oplus g(V_k)$$ are just a permutation of $$V_1, \dots, V_k$$. That is, $$g(V_i)= V_{\sigma(i)}$$. If no such decomposition is possible then we say that $$G$$ is primitive.

No. Consider $$\zeta_3 I \times A_5$$ a 180 element group given by a direct product of the $$A_5$$ subgroup of $$SO_3(\mathbb{R})$$ with a $$\zeta_3$$ global phase, as a subgroup of $$SU_3$$. This group is self normalizing but not maximal. Indeed it is contained in $$\zeta_3 I \times SO_{3}(\mathbb{R})$$ In general, any group which is maximal among the finite subgroups of $$SU_n$$ but is not Lie primitive (Lie primitive means not contained in any proper positive dimensional subgroup of $$SU_n$$) will be an example of a primitive self normalizing group which is not maximal among the closed subgroups of $$SU_n$$. Another example of this type would be $$N(SL(2,5)\otimes SL(2,5))$$ a 28,800 element maximal finite subgroup of $$SU_4$$ which is not Lie primitive since it is contained in $$N(SU_2 \otimes SU_2)\cong N(SO_4(\mathbb{R}))$$ Another example is $$N(2.S_6)$$ a 2,880 element maximal finite subgroup of $$SU_4$$ which is not Lie primitive since it is contained in $$N(Sp_2)$$