Some Characterizations of a Normal Subgroup of a Group and Isotopic Classes of Transversals

Abstract

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this note, we give some characterizations for normality of H in G. As a consequence we get a very short and elementary proof of the Main Theorem of [5], which avoids the use of the classification of finite simple groups.

Isotopic Classes of Transversals in Dihedral Group D2n, n Odd

“…In Section 2, we describe the number of elements in Itp(D 2n , H), where n is an odd integer greater than 1 and H is a subgroup of order 2 of the dihedral group D 2n of order 2n. The results proved are generalizations of results in the Section 4 of [5]. In Section 3, we compute the cyclic index of the one dimensional affine group Aff(1, p 2 ) of Z p 2 , where p is an odd prime .…”

“…It can be easily verified that (Z n , • A ) is a right loop (see [5,Section 4]). We denote this right loop by Z A n .…”

“…Then Z A n = Z n . Since T (G, H) contains exactly one loop transversal (see [5,Corollary 4.5]) and a right loop isotopic to a loop is itself a loop (see [5,Corollary 3.2]), we are done in this case. Next, assume that A = φ.…”

“…Let S denote a permutation group on a finite set S. [2, p.146]). Now we state the following theorem whose proof may be imitated from the proof of the Theorem 4.9 of [5] replacing [5,Theorem 4.7] by Theorem 2.1, Z p by Z n , and Aff(1, p) by Aff(1, n). Theorem 2.2.…”