ON B(5,k) GROUPS

Abstract

A group G is said to be a B(n, k) group if for any n-element subset A of G, |A 2 | ≤ k. In this paper, characterizations of B(5, 16) groups and B(5, 17) groups are given.2010 Mathematics subject classification: primary 20D60; secondary 20E34. Keywords and phrases: small squaring property, B(n, k) groups. IntroductionA group G is said to have the small squaring property on n-element subsets if for any n-element subset A of G, |A 2 | < n 2 , where A 2 = {ab | a, b ∈ A}. Furthermore, G is called a B n -group if |A 2 | ≤ 1 2 n(n + 1) for all n-element subsets A. Recently, Eddy and Parmenter generalized these notions to a new notion of B(n, k) groups [3]. A group G is called a B(n, k) group if |A| = n implies |A 2 | ≤ k with k ≤ n 2 − 1. Therefore a B n -group is a B(n, 1 2 n(n + 1)) group, and a group with small squaring property on n-element subsets is a B(n, n 2 − 1) group. Determining all B(n, k) groups is an interesting problem. For any given k, G is obviously a B(n, k) group when |G| ≤ k, and such G is referred to as trivial. It is also easy to see that any abelian group G is a B(n, k) group when k ≥ 1 2 n(n + 1). So what we are interested in is to determine all nonabelian nontrivial B(n, k) groups.The B(n, k) groups for n = 2 and n = 3 have been completely characterized in the literature [1,3, 4,9,10]. For n = 4, all B(4, 10) groups were characterized by Parmenter in [10], and B(4, k) groups where k = 11, 12, and 13 were recently characterized by Li and Tan in [7,8]. The only known result for B(5, k) groups with k ≥ 15 is the characterization of B(5, 15) groups given by Li and Tan in [6], and it was shown that G is a nonabelian nontrivial B(5, 15) group if and only if G Q 8 × C 2 . In this paper, we continue the investigation on B(5, k) groups for k = 16 and 17. We provide the complete characterizations of B(5, 17) non-2-groups and 2-groups in Sections 2 and 3, respectively. In Section 4 we obtain a complete characterization of B(5, 16) groups, and we also give a short proof for the characterization of B(5, 15) groups.Throughout the paper, all nonabelian groups are assumed to be finite, and our notation for groups is standard and follows that in [11]. In particular, we denote the quaternion group of order eight and the dihedral group of order 2n by Q 8 and D 2n , respectively: The characterization of B(5, 17) non-2-groupsIn this section, we investigate B(5, 17) non-2-groups. We first work on a necessary condition for a non-2-group G to be a B(5, 17) group. Afterwards, we will give a complete characterization of B(5, 17) non-2-groups. Throughout the section, a group G is assumed to be a non-2-group.2.1. A necessary condition for B(5, 17) non-2-groups. We first characterize a Sylow subgroup of odd order of a B(5, 17) group.L 2.1. Let P be a Sylow subgroup of odd order of a B(5, 17) group G. Then P is abelian.P. Suppose on the contrary that P is not abelian. Then P has two maximal subgroups M andSince M and N are maximal subgroups of P, M P and N P. Since N, aN and a 2 N are disjoint, we may write...

Theoretical studies of reactions of carbon dioxide mediated and catalysed by transition metal complexes

“…In 2011, Jun and Yoshizawa studied theoretically insertion reaction of CO 2 with the 18-electron complex (PNP)IrH 3 (PNP = 2,6-(diisopropylphosphinomethyl)-pyridine). 28 Ligand dissociation is difficult to occur Scheme 6…”

“…The reaction mechanism was also extensively investigated theoretically. 28,30,92 The proposed mechanism on the basis of these theoretical studies can be summarized as in Scheme 32. Starting with a trihydride complex, CO 2 inserts into an Ir-H bond to give an iridium formate via a direct hydrogen abstraction process (Section 2.5).…”

On B(n, k) 2-groups

“…For n = 4, all the B 4 10 groups were characterized by Parmenter in [13], and the B 4 k groups for k = 11 12 and 13 were characterized in [9,10]. For the case when n = 5, the B 5 k groups were characterized in [6][7][8] for 15 ≤ k ≤ 18; one particularly interesting result in these articles is that there are finitely many nonabelian B 5 k groups for 15 ≤ k ≤ 18. The process used in these articles to determine if groups of small order are B 5 k 2-groups was long due to hand calculation.…”

OnB(4, 13) 2-Groups

“…The B n k groups for n = 2 and n = 3 have been completely characterized in the literature [2-4, 6, 7]. Recently, all B 4 10 groups and all B 5 15 groups were characterized by Parmenter and the authors, respectively, [7,8]. Most recently, the characterizations of B 4 11 groups, B 4 12 groups, and B 4 13 non-2-groups were obtained by the authors [9].…”