On Connected Transversals to Abelian Subgroups in Finite Groups

On Connected Transversals to Subgroups Whose Order is a Product of Two Primes

“…If a H = bH and b −1 a / ∈ Q then again G is generated by four elements from A ∪ B and |G| ≤ p 4 q 8 . Thus we may assume that b −1 a ∈ Q whenever a H = bH and we are in a situation which is similar to the one described in the proof of Theorem 3.1 of [5]. We proceed as in that proof and we conclude that A = B.…”

“…If the elements h(a, b) ∈ Q for all a, b ∈ A then we can proceed as in the proof of Theorem 3.1 in [5] and we conclude that A is an abelian group. Thus G = AH and it follows from Lemma 2.5 that G is soluble.…”

“…Let G be a finite group and H a subgroup of G. If A and B are two left transversals to H in G and [A, B] ≤ H then we say that A and B are H -connected in G. This concept was introduced in [5] and it was used to characterize multiplication groups of loops. We then started to investigate the relation between the solubility of G and the structure of H .…”

“…If H is a subgroup of a group G then H G denotes the core of H in G (the largest normal subgroup of G contained in H ). Basic facts about connected transversals, loops and their multiplication groups can be found in [3,5,7]. In this paper we consider finite loops and groups only.…”

Bruck Loops with Abelian Inner Mapping Groups

“…Much work in loop theory was devoted to attempting to prove this [9,16,24]. However, in 2004, Csörgő [7] constructed a loop Q of order 128, with abelian inner mapping group, and with c (Q) = 3.…”