Clifford Theory - Proof of Clifford's Theorem

Proof of Clifford's Theorem

The proof of Clifford's theorem is best explained in terms of modules (and the module-theoretic version works for irreducible modular representations). Let F be a field, V be an irreducible F-module, V_N be its restriction to N and U be an irreducible F-submodule of V_N. For each g in G, U.g is an irreducible F-submodule of V_N, and is an F-submodule of V, so must be all of V by irreducibility. Now V_N is expressed as a sum of irreducible submodules, and this expression may be refined to a direct sum. The proof of the character-theoretic statement of the theorem may now be completed in the case F = C. Let χ be the character of G afforded by V and μ be the character of N afforded by U. For each g in G, the C-submodule U.g affords the character μ(g) and . The respective equalities follow because χ is a class-function of G and N is a normal subgroup. The integer e appearing in the statement of the theorem is this common multiplicity.