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Induction-isotypical subgroup
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Template:Rep-theoretic subgroup property
Contents |
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 σ of H, the induction of σ 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.
