Generalized Dedekind's theorem and its application to integer group determinants

Abstract

In this paper, we give a refinement of a generalized Dedekind's theorem, which is a generalization of Laquer's theorem. In addition, we show that all possible values of an integer group determinant are also possible values of the integer group determinant of any its abelian subgroup.

Integer circulant determinants of order 16

“…Lemma 5.4. The following hold: We can obtain the same conclusion for the case (iv) by using Lemma 2.10 (4) and (5).…”

“…, y 15 ) := det (y gh −1 ) g,h∈C 2 4 . From the G = C 4 and H = {0, 2} case of [5,Theorem 1.1], we have the following corollary.…”

“…For every group G of order at most 15, S(G) was determined (see [3,4]). Let C n be the cyclic group of order n. For the groups of order 16, the complete descriptions of S(G) were obtained for the dihedral group [1,Theorem 5.3], the cyclic group [7] and the direct groups C 4 2 [8] and C 8 × C 2 [5,Theorem 1.5]. In this paper, we determine S (C 2 4 ).…”

“…The problem suggested by Olga Taussky-Todd is extended to the problem that is to determine S(G) for any finite group G. For some groups, the problem was solved in [1,2,3,5,7,8,9,11,13,14,15]. As a result, for every group G of order at most 15, S(G) was determined.…”

“…Remark that C ⊂ D holds. In [14], [15], [11,Theorem 1.5], [1,Theorem 5.3] and [13], the following are obtained respectively:…”