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Let G be a monomial group, and let H_1,...,H_r be the subgroups of G where there exists a linear character that induces to an irreducible character of G. How much is known about these subgroups? For example: What can be said about their intersection, which ones are normal, etc?

I ask because I have found some results related to this, and I want to know what there exists of current theory on this topic.

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One nice theorem I know in this direction is a beautiful result of U. Riese which states that if $A$ is an Abelian subgroup of $G$ such that ${\rm Ind}_{A}^{G}(\lambda)$ is irreducible for some linear character $\lambda$ of $A$, then $A \lhd \lhd G.$

Riese, Udo, A subnormality criterion in finite groups related to character degrees, J. Algebra 201, No. 2, 357-362 (1998). ZBL0915.20004.

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