Induction-isotypical subgroup

Definition

Definition with symbols

Let $H$ be a subgroup of a group $G$ and let $k$ be a field. We say that $H$ is induction-isotypical in $G$ with respect to $k$, if, for every irreducible representation $\sigma$ of $H$, the induction of $\sigma$ to $G$ is an isotypical representation of $G$.

Relation with other properties

When the field is sufficiently large

In case of a sufficiently large field for the group, the subgroup property of being induction-isotypical is equivalent to the subgroup property of being a conjugacy-closed normal subgroup. This follows as a consequence of the conjugacy class-character duality.

In the general non-modular case

In the general non-modular case, the subgroup property of being induction-isotypical is equivalent to the subgroup property of being a normal subgroup with the property that all Galois-class automorphisms restrict to Galois-class automorphisms (also called a Galois-conjugacy-closed normal subgroup).

In the general (possibly modular) case

In the modular case, it is no longer true that the character determines the representation. Hence, being conjugacy-closed may not suffice for being induction-isotypical. However, every central factor is always induction-isotypical regardless of the field under consideration.