The minimal degree for a class of finite complex reflection groups

Abstract

We calculate the minimal faithful permutation degree for a class of finite complex reflection groups G(p, p, q), for p and q primes, and demonstrate that they form examples where the minimal degree of a direct product is strictly less than the sum of the minimal degrees of the direct factors.

The minimal degree of permutation representations of finite groups

“…It should be noted that there are pairs of groups G and H such that µ(G × H) < µ(G) + µ(H). For examples, see [2] or [3]. It should be also noted that in [2], after proving the formula for nilpotent groups, the same formula is proved for an extended collection of groups, each containing a "large enough" nilpotent subgroup.…”

Minimal faithful permutation degrees for irreducible Coxeter groups and binary polyhedral groups

“…Here, the group G could be taken to be G (2,2,5) and H to be the centraliser of G in Sym (10) which is cyclic of order 2. Moreover in [8,9], the author studied the minimal degrees of the complex reflection groups G(p, p, q), where p and q are primes and constructed an infinite class of examples where strict inequality holds (see Section 8 for a full definition of this family of groups). It was always the case that for G = G(p, p, q), where p and q are distinct primes satisfying certain other conditions, we have µ(G) = µ(G × C Sym(µ(G)) (G))…”

“…. , G l }, where l ≥ 1 and the G i are the point stabilisers of the orbits of G, we may express µ(G) as the smallest value of l i=1 |G : G i | such that the intersection of the cores of the G i is trivial; recall core(G i ) = g∈G G g i (see [4], [9]). It is common to call a collection of subgroups which furnishes µ(G) the minimal representation (though it may not in general be unique).…”

The Smallest Faithful Permutation Degree for a Direct Product Obeying an Inequality Condition

“…In that example, G and H generate a subgroup GH of Sym(15) that is an internal direct product of G and H. Saunders showed in [6] that the example in [8] fits into a general family that provides infinitely many instances of strict inequality in (1). There G could be taken to be the complex reflection group G(p, p, q), where p and q are distinct odd primes satisfying certain other conditions, and H the centraliser of the minimally embedded image of G in Sym(pq).…”