On the non-abelian tensor square of all groups of order dividing p5

In this paper, we consider all groups of order dividing [Formula: see text]. We obtain the explicit structure of the non-abelian tensor square, non-abelian exterior square, tensor center, exterior center, the third homotopy group of suspension of an Eilenberg–MacLane space [Formula: see text] and [Formula: see text] of such groups.

Rankin–Selberg L-functions via good sections

In this article, we revisit Rankin–Selberg integrals established by Jacquet, Piatetski-Shapiro and Shalika. We prove the equality of Rankin–Selberg local factors defined with Schwartz–Bruhat functions and the factors attached to good sections, introduced by Piatetski-Shapiro and Rallis. Moreover, we propose a notion of exceptional poles in the framework of good sections. For cases of Rankin–Selberg, Asai and exterior square L-functions, the exceptional poles are consistent with well-known exceptional poles which characterize certain distinguished representations.

The Exterior Square of a Bieberbach Group with Quaternion Point Group of Order Eight

A Bieberbach group is defined to be a torsion free crystallographic group which is an extension of a free abelian lattice group by a finite point group. This paper aims to determine a mathematical representation of a Bieberbach group with quaternion point group of order eight. Such mathematical representation is the exterior square. Mathematical method from representation theory is used to find the exterior square of this group. The exterior square of this group is found to be nonabelian. Keywords: mathematical structure; exterior square; Bieberbach group; quaternion point group

ON -GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE CHEVALLEY GROUP

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\text{Aut}(P)$ induces $N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$ on the Frattini quotient $P/\unicode[STIX]{x1D6F7}(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p=3$ our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_{2}(q)$ on the octonion algebra over $\mathbb{F}_{q}$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_{q}[G_{2}(q)]$-module.

A polycyclic presentation for the 𝑞-tensor square of a polycyclic group

Let G be a group and q a non-negative integer. We denote by {\nu^{q}(G)} a certain extension of the q-tensor square {G\otimes^{q}G} by {G\times G}. In this paper, we describe an algorithm for deriving a polycyclic presentation for {G\otimes^{q}G} when G is polycyclic, via its embedding into {\nu^{q}(G)}. Furthermore, we derive polycyclic presentations for the q-exterior square {G\wedge^{q}G} and for the second homology group {H_{2}(G,\mathbb{Z}_{q})}. Additionally, we establish a criterion for computing the q-exterior center {Z_{q}^{\wedge}(G)} of a polycyclic group G, which is helpful for deciding whether or not G is capable modulo q. These results extend to all {q\geq 0} generalizing methods due to Eick and Nickel for the case {q=0}.

The Schur multiplier of groups of order 𝑝5

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian p-groups of order {p^{5}}. As an application, we determine the capability of groups of order {p^{5}}.