Monomial group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

Symbol-free definition

A finite group is termed monomial (or sometimes, a M-group or M_1-group) with respect to a field k (whose characteristic does not divide the group order) if it satisfies the following equivalent conditions:

  1. Every irreducible representation of the group over k is induced from a one-dimensional representation of a subgroup, i.e., a linear character.
  2. Every finite-dimensional linear representation of the group over k is a monomial linear representation: it is a direct sum of representations induced from one-dimensional representations of subgroups.

Relation with other properties

Stronger properties

Weaker properties