Isotopic Classes of Transversals in Dihedral Group D 2n , n Odd
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RESEARCH ARTICLE
Isotopic Classes of Transversals in Dihedral Group D2n, n Odd Surendra Kumar Mishra1 • R. P. Shukla1
Received: 4 September 2018 / Revised: 12 July 2019 / Accepted: 12 July 2019 Ó The National Academy of Sciences, India 2019
Abstract In this paper, we determine the number of isotopic classes of transversals of a subgroup of order 2 in D2n (n is a positive odd integer greater than 1), where isotopism classes are formed with respect to the induced right loop structures. We also determine the cyclic index of the affine group Affð1; p2 Þ, where p is an odd prime. Keywords Transversals Right loop Isotopy Cyclic index Mathematics Subject Classification 20D60 20N05
1 Introduction Let G be a finite group and H be a subgroup of G. A normalized right transversal (NRT) T of H in G is a collection of elements of G obtained by selecting one and only one element from each right coset of H in G and 1 2 T. Let T be an NRT of H in G. Then T has a binary operation induced by the binary operation of G, given by fx yg ¼ Hxy \ T with respect to which T becomes a right loop with identity 1, that is a right quasigroup with both sided identity (see [1, Proposition 4.3.3, p. 102] and [2]). Conversely, every right loop is embedded as an NRT in a group with some universal property [2, Theorem 3.4].
& Surendra Kumar Mishra [email protected] R. P. Shukla [email protected] 1
Let ðL1 ; 1 Þ and ðL2 ; 2 Þ be two groupoids. We say that L1 is isotopic to L2 if there are bijective maps f, g and h from L1 to L2 such that f ðaÞ 2 gðbÞ ¼ hða 1 bÞ for all a; b 2 L1 . This type of triple (f, g, h) is known as an isotopism or an isotopy from L1 to L2 . An isotopy (f, f, f) from L1 to L2 is known as an isomorphism. We say that ðL1 ; 1 Þ and ðL1 ; 2 Þ are principal isotopic if (f, g, I) is an isotopy between ðL1 ; 1 Þ and ðL1 ; 2 Þ, where I is the identity map on L1 [3]. Let T ðG; HÞ denote the set of all NRTs to H in G. Let T1 ; T2 2 T ðG; HÞ. If the induced right loop structures in T1 and T2 are isotopic, then we say that T1 and T2 are isotopic. We denote the set of all isotopism classes of elements in T ðG; HÞ by I tpðG; HÞ. Let ðS; Þ be a right loop. For a 2 S, define La : S ! S by La ðxÞ ¼ a x and Ra : S ! S by Ra ðxÞ ¼ x a. An element a 2 S is said to be a left nonsingular if La is a bijection of S. It is observed [3, Theorem 1 A] that if a 2 S is left nonsingular and if b 2 S, then there is a principal isotopy ððRb Þ1 ; ðLa Þ1 ; IÞ between ðS; a;b Þ and ðS; Þ, where for all a; b 2 S, a a;b b ¼ ðRb Þ1 ðaÞ ðLa Þ1 ðbÞ and every principal isotope of ðS; Þ is of this form. We denote this isotope of S by Sa;b . It is easy to observed that the identity of ðS; a;bÞ is a b . It is also observed [3, Lemma 1 A] that if a right loop ðL1 ; 1 Þ is isotopic to the right loop ðL2 ; 2 Þ, then ðL2 ; 2 Þ is isomorphic to a principal isotope of ðL1 ; 1 Þ.
2 Isotopic Classes of Transversals Let n 2 N. Consider Zn ¼ f0; 1; 2; . . .; n 1g the ring of integers modulo n. Let A Zn nf0g. Le
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