Clifford theory

Remarks

For the proof of part 1, take an irreducible $N$-subrepresentation $W$ of $V$. Each $gW$ is again $N$-invariant and irreducible, and isomorphic to a conjugate of $W$. The sum $\sum_g gW$ is a nonzero $G$-invariant subspace, so it is $V$. The elements of $G$ permute the isotypic components of $V|_N$ transitively, and $T$ is the stabiliser of the component $U$. Part 2 follows, since a representation that is a direct sum of transitively permuted subspaces is induced from a stabiliser. Example: let $G = S_3$, $N = A_3 \cong \Z/3$ and let $V$ be the irreducible complex representation of degree $2$. Then $V|_N$ is the sum of the two non-trivial characters of $A_3$, which conjugation by a transposition swaps. Here $e = 1$, $t = 2$, $T = N$, and $V = \operatorname{Ind}_{A_3}^{S_3} \omega$ for a non-trivial character $\omega$ of $A_3$. The theorem also holds when $G$ is any group and $N$ has finite index in $G$.

Going deeper

For a student

This is the basic tool for building representations of a group from those of a normal subgroup and of the quotient. It applies well to solvable groups, which have chains of normal subgroups. In character language it states that the irreducible constituents of $\chi|_N$ form one $G$-orbit and all occur with the same multiplicity $e$. The Clifford correspondence, part 2, gives a bijection between irreducible characters of $T$ lying over $W$ and irreducible characters of $G$ lying over $W$, by induction. It is a standard step in inductive arguments in character theory, which pass from $G$ to a normal subgroup.