On the nonabelian tensor square and capability of groups of order p 2 q

Abstract

A group G is said to be capable if it is isomorphic to the central factor group H/Z(H) for some group H.

On the non-abelian tensor square of groups of order p4 where p is an odd prime

“…We know that K is isomorphic to non-abelian group of order p 3 of exponent p 2 . Again by using Theorem 5, we have G 5 ⊗ G 5 ∼ = (K × z ) ⊗ (K × z ) ∼ = (Z p ) 9 . Lastly, for group G 6 , we have G 6 = Z p and G…”

On the capability of nonabelian groups of order p4

“…In 2012, the capability of groups of order 2 p q has been computed by Samad et al in [5]. Recently, Samad et al in [6] determined the capability of groups of order 3 , p q where p and q are distinct primes and p < q.…”

Groups, Algorithms and Programming (GAP) and the Nonabelian Tensor Square of Groups of Order 8q

“…Hannebauer [3] determined the nonabelian tensor square of SL (2, q), PSL (2, q), GL (2, q) and PGL (2, q) for all q ≥ 5 and q = 9. The Schur multiplier and nonabelian tensor square of the nonabelian groups of order p 2 q, groups of order p 3 q and special orthogonal groups have been computed by Rashid et al in [4,5,6,7,8,9], where p and q are distinct primes.…”

“…where α is a primitive root of α 4 ≡ 1(mod q), 4 divides q − 1, <a, b, c| a 4 =b 2 =c q = 1, ab=ba, a -1 ca=c α , bc=cb> (1.1.11) where α is a primitive root of α 4…”