Clifford Theory - Corollary of Clifford's Theorem

Corollary of Clifford's Theorem

A corollary of Clifford's theorem, which is often exploited, is that the irreducible character χ appearing in the theorem is induced from an irreducible character of the inertial subgroup I_G(μ). If, for example, the irreducible character χ is primitive (that is, χ is not induced from any proper subgroup of G), then G = I_G(μ) and χ_N = eμ. A case where this property of primitive characters is used particularly frequently is when N is Abelian and χ is faithful (that is, its kernel contains just the identity element). In that case, μ is linear, N is represented by scalar matrices in any representation affording character χ and N is thus contained in the center of G (that is, the subgroup of G consisting of those elements which themselves commute with every element of G). For example, if G is the symmetric group S₄, then G has a faithful complex irreducible character χ of degree 3. There is an Abelian normal subgroup N of order 4 (a Klein 4-subgroup) which is not contained in the center of G. Hence χ is induced from a character of a proper subgroup of G containing N. The only possibility is that χ is induced from a linear character of a Sylow 2-subgroup of G.