Computing the nonabelian tensor squares of groups of order $$\pmb {p^3q}$$

Abstract

In this paper, we describe the explicit structures of nonabelian tensor squares of nonabelian groups of order $$p^3q$$ p 3 q , where p and q are distinct prime numbers and $$p>2$$ p > 2 . Our method is based on determining the structures of their Schur multipliers by applying some well-known results and the presentations of groups, which leads us to obtain the orders of their nonabelian tensor squares.

Capable groups of order p3q

“…where α is any primitive root of α p ≡ 1 (mod q). It follows from [11] that G ∧ G ∼ = G ′ ∼ = C pq . One could readily seen that Z ∧ (G) = Z(G) which implies that G is unicentral.…”

“…In the case M(G) ∼ = C p , őrst observe by [11,Corollary 3.4] that |G ⊗ G| = p 4 q. Also, [11,Proposition 2.4] and exact sequence…”

“…In order to study the capability of groups of order p 3 q, p > 2, we őrst compute their nonabelian exterior squares. Also it will be useful to know the nonabelian tensor squares and Schur multipliers of such groups which are described in [11]. The following result (part (ii)) will be used in proof of Proposition 3.…”

“…Then from [11,Proposition 3.2] we have M(G) ∼ = 1, C p , C 2 p or C 3 p . By a same argument as for the group (18) we can show that G ∧ G ∼ = C 3 p , Φ 2 (1 4 ), Φ 4 (1 5 ) or Φ 11 (1 6 ), respectively (for more details see the proof of Theorem B in [11]). Note that the polycyclic presentation of G is as follows:…”

“…Let G be a nonabelian group of order p 3 q, where p and q are distinct prime numbers and p > 2. Then G is capable if and only if G is any group of types ( 6), ( 8), (10), (11), ( 12), ( 15), ( 16), (18), ( 19), (21), ( 22) or (25).…”