INDUCED CYCLE STRUCTURES OF THE HYPEROCTAHEDRAL GROUP

In this paper, the n-dimensional hypercube Q(n) is treated as a graph whose vertex set consists of sequences of 0's and 1's of length n, and the hyperoctahedral group B(n) is the automorphism group of Q(n). It is well known that B(n) can be represented by the group of signed permutations, namely, any signed permutation induces a permutation on the vertices of Q(n), which preserves adjacency. Moreover, the set of signed permutations on n elements also induces a permutation group on the edges of Q(n), denoted H(n). The author studies the cycle structures of both B(n) and H(n). The method proposed here is to determine the induced cycle structure by computing the number of fixed vertices or fixed edges of a signed permutation in the cyclic group generated by a signed permutation of given type. Here we define the type of a signed permutation by a double partition based on its signed cycle decomposition. In this way, one can compute the cycle indices of both B(n), and H(n) by counting fixed vertices and fixed edges of a signed permutation. The formula for the cycle index of B, is much more natural and considerably simpler than that of Harrison and High [J. Combin. Theory, 4 (1968), pp. 277-299]. Meanwhile. the cycle structure of H(n) seems not to have been studied before, although it is well motivated by nonisomorphic edge colorings of Q(n), as well as by the recent interest in edge symmetries of computer networks.