In the survey article [2] it was noted, among many other open problems, that the classification of the groups acting regularly on a d-dimensional cube Γ is unsettled. In other words, the classification of the finite groups G such that Cay(G, S) ≅ Γ, for some subset S of G, is still unknown. In this article, we prove that there are at least 2d2/64-(d/2)log2(d/2) nonisomorphic 2-groups of Frattini class 2 acting regularly on a d-dimensional cube. Other relevant results are presented. As a corollary of our result, we remark that the symmetric group Sym(n) on n symbols contains at least 2n2/256-(n/4)log2(n/4) subgroups up to isomorphism. In particular, we recall that in [4] it was proved that the total number of subgroups of Sym(n) is at most 2cn2, for c = log224. © Taylor & Francis Group, LLC
Spiga, P. (2009). Enumerating groups acting regularly on a d-dimensional cube. COMMUNICATIONS IN ALGEBRA, 37(7), 2540-2545 [10.1080/00927870902766308].
Enumerating groups acting regularly on a d-dimensional cube
SPIGA, PABLO
2009
Abstract
In the survey article [2] it was noted, among many other open problems, that the classification of the groups acting regularly on a d-dimensional cube Γ is unsettled. In other words, the classification of the finite groups G such that Cay(G, S) ≅ Γ, for some subset S of G, is still unknown. In this article, we prove that there are at least 2d2/64-(d/2)log2(d/2) nonisomorphic 2-groups of Frattini class 2 acting regularly on a d-dimensional cube. Other relevant results are presented. As a corollary of our result, we remark that the symmetric group Sym(n) on n symbols contains at least 2n2/256-(n/4)log2(n/4) subgroups up to isomorphism. In particular, we recall that in [4] it was proved that the total number of subgroups of Sym(n) is at most 2cn2, for c = log224. © Taylor & Francis Group, LLC
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