Some Notes on Higher Frobenius-Schur Indicators of the Regular Representations for Matched Pairs of Groups

The notion of matched pair (F, G, (sic), (sic)) of finite groups was introduced by Takeuchi in 1981, which is equivalent to a factorization of the group F (sic) G. We find in this paper some sufficient conditions when equations nu(m)(F (sic) G) = nu(m)(F)nu(m)(G) for all m >= 0 imply that F (sic) G is the external direct product of F x G, where nu(m) denotes m-th indicator of the regular representation of a finite group. A comparison with indicators of bismash product Hopf algebras C-G#CF is also mentioned.