Related papers: The Multiplicative Jordan Decomposition in the Int…

Determining the Jordan canonical form of the tensor product of Jordan blocks has many applications including to the representation theory of algebraic groups, and to tilting modules. Although there are several algorithms for computing this…

For finite p-groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P: the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of…

Jespers and Sun conjectured that if a finite group $G$ has the property ND, i.e. for every nilpotent element $n$ in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in…

Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively…

We obtain restrictions on units of even order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ by studying their actions on the reductions modulo $4$ of lattices over the $2$-adic group ring $\mathbb{Z}_2G$. This improves the…

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[\zeta_m+\zeta_m^{-1}])$ and $K_{2k}(\mathbb{Z}[\zeta_m])$,…

For every prime $p>3$ we prove that $Q \times \mathbb{Z}_p$ and $\mathbb{Z}_2^3 \times \mathbb{Z}_p$ are DCI- groups. This result completes the description of CI-groups of order $8p$.

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces…

For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the…

Let p be a prime number and G be a finite commutative group such that p^{2} does not divide the order of G. In this note we prove that for every finite module M over the group ring Z_{p}[G], the inequality #M \leq…

Let $RG$ be the group ring of an arbitrary group $G$ over an associative non-commutative ring $R$ with identity. In this paper, we have obtained the necessary and sufficient conditions under which $RG$ is Jordan nilpotent of index $4$.

Let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field $F$ of characteristic $p$. For positive integers $r$ and $s$ with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes…

We study an integer sequence associated with Cantor's division polynomials of a genus 2 curve having an integral point. We show that the reduction modulo $p$ of such a sequence is periodic for all but finitely many primes $p$, and describe…

If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv…

Let $\Bbb Z$ be the set of integers, and let $p$ be a prime of the form $4k+1$. Suppose $q\in\Bbb Z$, $2\nmid q$, $p\nmid q$, $p=c^2+d^2$, $c,d\in\Bbb Z$ and $c\equiv 1\pmod 4$. In this paper we continue to discuss congruences for…

Let $J_r$ denote an $r\times r$ matrix over a finite field $F$ with minimal and characteristic polynomials $(t-1)^r$. Suppose $r\leq s$. It is not hard to show that the Jordan canonical form of $J_r\otimes J_s$ is similar to…

For a prime $p\equiv 3$ $(\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\mathbb{Q}(\sqrt{-2p})$ and $\mathbb{Q}(\sqrt{2p})$, respectively. Let $\Psi(\xi)$ be the Hirzebruch sum of a quadratic irrational $\xi$. We show…

In this article, we determine all inequivalent irreducible rational matrix representations of groups of order $p^5$, where $p$ is an odd prime. We also derive combinatorial formulations for the Wedderburn decomposition of rational group…

In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials…

We classify all $n$-dimensional reduced Cohen-Macaulay modular quotient variety $\mathbb{A}_\mathbb{F}^n/C_{2p}$ and study their singularities, where $p$ is a prime number and $C_{2p}$ denotes the cyclic group of order $2p$. In particular,…