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A Note on Orbits of Subgroups of the Permutation Groups

In [2] we studied Milgram's Complex, C(n – 1), which was first defined in [3], in the following manner. Let Sn , the permutation group of n symbols, act on R n in the obvious manner; put α(x) = (y), where yi = xα –1 (i) . Let s = (1, 2, …, n), then C(n – 1) is the convex hull of the points α(s), α ∊ Sn . Here we shall generalise this construction as follows. Let G be a subgroup of Sn , and let v ∊ R n . Then C(G, v) is the convex hull of α(v), α ∊ G. We prove invariance over v subject to certain restrictions, give counter-examples to shew lack of invariance if we alter G, discuss how we may describe C(G, v), shew that the only “nice” case is essentially when G is Sn , and lastly give some examples.