In this paper and its sequel we study arrays of coupled identical cells that possess a ‘global’ symmetry group G, and in which the cells possess their own ‘internal’ symmetry group L. We focus on general existence conditions for symmetry-breaking steady-state and Hopf bifurcations. The global and internal symmetries can combine in two quite different ways, depending on how the internal symmetri...

Given natural numbers m and n, we define a deflation map from the characters of the symmetric group Smn to the characters of Sn. This map is defined by first restricting a character of Smn to the wreath product Sm oSn, and then taking the sum of the irreducible constituents of the restricted character on which the base group Sm × · · · × Sm acts trivially. We prove a combinatorial formula which...

In 1954 A. G. Howson proved that the intersection of two finitely generated subgroups of a free group is again finitely generated. Now the free metabelian subgroups of a free metabelian group of finite rank n are quite restricted. Indeed they are again of finite rank at most n. This suggests that there may be an analog of Howson’s theorem for free metabelian groups. This turns out not to be the...