Abstract
For each finite group G, the product in the group ring of all the conjugacy class sums is a positive integer multiple of the sum of the elements in a special coset of the commutator subgroup G′, as Brauer and Wielandt first observed in the case G′ = G. We show that the corresponding special element G! in A := G/G′ is the product of B! over specified subgroups B of A. Somewhat analogously, the product of all the irreducible characters of G, restricted to the center Z of G, is a multiple of a special linear character !G of Z, and !G is the product of !(Z/Y) over specified subgroups Y of Z.
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Gallagher, P.X. Class product and character product. Arch. Math. 102, 201–207 (2014). https://doi.org/10.1007/s00013-014-0625-5
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DOI: https://doi.org/10.1007/s00013-014-0625-5