Degree homogeneous subgroups

Let G be a finite group and H be a subgroup. We say that H is degree homogeneous if, for each χ ∈ Irr(G), all the irreducible constituents of the restriction χH have the same degrees. Subgroups which are either normal or abelian are obvious examples of degree homogeneous subgroups. Following a question of E.M. Zhmud’, we investigate general properties of such subgroups. It appears unlikely that degree homogeneous subgroups can be characterized entirely by abstract group properties, but we provide mixed criteria (involving both group structure and character properties) which are both necessary and sufficient. For example, H is degree homogeneous in G if and only if the derived subgroup H ′ is normal in G and, for every pair α, β of irreducible G-conjugate characters of H ′, all irreducible constituents of α and β have the same degree. Mathematics Subject Classification (2000): 20C15 ∗The first author acknowledges partial support from NSERC Operating Grant A7171. The second author’s visit to Carleton University was made possible through this grant and he also thanks Carleton University for its hospitality during the time that this paper was written.