International Journal of Group Theory
Volume:5 Issue: 4, Dec 2016

In this paper, we will give necessary and sufficient conditions for certain HNN extensions of subgroup separable groups with normal associated subgroup to be conjugacy separable. In fact, we will show that these HNN extensions are conjugacy separable if and only if the normalizer of one of its associated subgroup is conjugacy separable.

In this paper we study the coprime graph of a group G G . The coprime graph of a group G G , is a graph whose vertices are elements of G G and two distinct vertices x x and y y are adjacent iff (|x|,|y|)=1 (|x|,|y|)=1 . In this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. Also we study the automorphism group of the coprime graph.

A positive solution of the problem of the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of sufficiently large odd periods n>10 10 n>1010 obtained previously by S. V. Ivanov and R. Mikhailov extended to all odd periods n≥665 n≥665 ý.

It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 256 (2009)] that the ZL amenability constant of a finite group is always at least~1 1 , with equality if and only if the group is abelian. It was also shown in [A. Azimifard, E. Samei, N. Spronk, op cit.] that for any finite non-abelian group this invariant is at least 301/300 301/300 , but the proof relies crucially on a deep result of D. A. Rider on norms of central idempotents in group algebras. Here we show that if G G is finite and non-abelian then its ZL-amenability constant is at least 7/4 7/4 , which is known to be best possible. We avoid use of Rider's reslt, by analyzing the cases where G G is just non-abelian, using calculations from [M. Alaghmandan, Y. Choi, E. Samei, CMB 57 2014)], and establishing a new estimate for groups with trivial centre.

In this paper, we generalize some transfer theorems.~In particular, we derive one of the main results of Gagola(Contemp Math 524:49--60, 2010) from our results.