Extending irreducible representations of normal subgroups

Abstract

Let H be a normal subgroup of some finite group G and let V be an irreducible (right) FH-module, where F is an arbitrary field. It is one of the primary objectives of Clifford theory to find necessary and sufhcient conditions so that V can be extended to an FG-module. An obvious necessary condition is that V has to be G-invariant (stable). Sufficiency relies upon various properties of the group extension H * G --w S, where S= G/H, and the module itself. The theory is well developed in case V is absolutely irreducible. Recently Isaacs [12] and Dade [4] proved, generalizing a theorem by Gallagher, that V can be always extended to an FG-module provided H is a Hall subgroup of G (and V is G-invariant). The present paper resulted from an effort to understand this remarkable and somewhat surprising result. Assume in the sequel that V is G-invariant. Let D = End& V)’ be the opposite endomorphism ring of K Basic Clifford theory yields a group extension D’ * 52 -P-P S such that there exists an FG-module extending V if and only if 52 =52,(V) splits over the multiplicative group D’ of the division algebra D (see Theorem 1 below). In general D is noncommutative and so the usual cohomological methods seem to fail. Dade even remarked in [4] sceptically that “none of the nice theorems about extensions of V, such as those of Gallagher, which depend upon calculations in the cohomology groups of S with coefficients in a commutative group, can be proved this way.” In the following situation, however, one can argue as usual: