The structure of groups with index-3 subgroups and Landau’s second theorem
Lawrence V. Meisel, D. M. Gray, E. Brown · Journal of Mathematical Physics · 1975
For any group G0 which contains an index-3 subgroup G, it is shown that either (a) G is an invariant subgroup or (b) G contains an index-2 subgroup GA, where GA is an invariant subgroup of G0. For case (a), G and its cosets give rise to three operators which span a stable three-dimensional subspace of the group algebra which further reduces to three one-dimensional stable subspaces. For case (b), GA and its cosets give rise to six operators which span a six-dimensional stable subspace of the group algebra which reduces to two one-dimensional and two two-dimensional irreducible stable subspaces of the group algebra. The irreducible representations and the corresponding basis elements of the group algebra are given for both cases. Landau’s second theorem pertaining to second order phase transitions is proven.