Showing papers in "Duke Mathematical Journal in 1972"

On the number of distinguished representations of a group element

Abstract: where a is an element of G and x, , , x, belong to D. Of course, if a is not in the subgroup generated by Dl then N,(a) = 0 for all r . In Proposition 1 we note that the evaluation of N t ( a ) reduces to the corresponding question for a certain quotient group of G. If D itself is a subgroup of GI then trivially N,(a) = JDIr-' for a in D. For arbitrary D the calculation of N t ( a ) seems quite difficult. One of our main results is the explicit determination in Theorem 2 of N:(a) when G \\ D , the complement of D in G, is a subgroup of G. As an application of Theorem 2 we obtain in Corollary 5 the number of representations of a given element in an abelian group G as a product of r elements of maximal order in G. In particular, for G cyclic this number agrees with a formula derived by Rearick [3] and is essentially equivalent to an earlier formula of Dixon [I]. In $4 analogous questions for rings are considered.