G be a finite group, C^× the non-zero complex numbers with trivial G-module structure. then H^2(G, C×) is finite.

Question

Let G be a finite group, C^× the non-zero complex numbers with trivial G-module structure. Show that H^2(G, C^×) is finite. H2(G/N, C*); C* is the G/N-trivial module of the group of nonzero complex numbers under multiplication. It is well known that H2(G/N, C*) = 0 when G/N is cyclic acting trivially on C*, a divisible group. Thus the projective representation associated to a: can be modified until it arises from an ordinary irreducible representation of the same degree. Since G/N is cyclic, we conclude that 01 must have degree 1. However, Clifford’s results also assert that OL has degree s. Thus s = 1 and (f) holds.

Answer 1

Let G be a finite group, C^× the non-zero complex numbers with trivial G-module structure. Show that H^2(G, C^×) is finite. H2(G/N, C*); C* is the G/N-trivial module of the group of nonzero complex numbers under multiplication. It is well known that H2(G/N, C*) = 0 when G/N is cyclic acting trivially on C*, a divisible group. Thus the projective representation associated to a: can be modified until it arises from an ordinary irreducible representation of the same degree. Since G/N is cyclic, we conclude that 01 must have degree 1. However, Clifford’s results also assert that OL has degree s. Thus s = 1 and (f) holds.