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We prove that if $G$ and $H$ are finite metacyclic groups with isomorphic rational group algebras and one of them is nilpotent then $G$ and $H$ are isomorphic.

The weights for a finite group G with respect to a prime number p where introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that that the number of G-conjugacy classes of weights of G coincides with the…

We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of…

Let $p$ be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite $p$-groups whose center has index $p^3$, which is a strong contrast to the analogous situation for $p = 2$.

Extending the results of [Asian J. Math. 2019], in [Doc. Math. \textbf{21}, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of \textit{odd} degree over the…

We study the problem of determining, for a polynomial function $f$ on a vector space $V$, the linear transformations $g$ of $V$ such that $f g = f$. In case $f$ is invariant under a simple algebraic group $G$ acting irreducibly on $V$, we…

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of…

We construct countable groups $G$ with the following new degree of W*-superrigidity: if $L(G)$ is virtually isomorphic, in the sense of admitting a bifinite bimodule, with any other group von Neumann algebra $L(\Lambda)$, then the groups…

In this paper, we address the following question: when is a finite $p$-group $G$ self-similar, i.e. when can $G$ be faithfully represented as a self-similar group of automorphisms of the $p$-adic tree? We show that, if $G$ is a self-similar…

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of…

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

In 2018, Legrand and Paran proved a weaker form of the Inverse Galois Problem for all Hilbertian fields and all finite groups: that is, there exist possibly non-Galois extensions over given Hilbertian base field with given finite group as…

A morphism of linear algebraic groups $\phi:K\rightarrow G$ is called an epimorphism if it admits right cancellation. A subgroup $H\leq G$ is epimorphic if the inclusion map is an epimorphism. For $G$ a simple algebraic group over an…

Finite $p$-groups with a unique $\mathcal{A}_2$-subgroup are classified up to isomorphism. A problem proposed by Berkovich and Janko is solved.

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group,…

We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to…

Let $G = H\times A$ be a group, where $H$ is a purely non-abelian subgroup of $G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n \geq 2$, we show that there exists an isomorphism $\phi : Aut_{Z(G)}^{\gamma_{n}(G)}(G)…

In recent work, Rosenbaum and Wagner showed that isomorphism of explicitly listed $p$-groups of order $n$ could be tested in $n^{\frac{1}{2}\log_p n + O(p)}$ time, roughly a square root of the classical bound. The $O(p)$ term is entirely…

We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of…

In two-dimensional conformal field theory, we analyze conformally invariant boundary conditions which break part of the bulk symmetries. When the subalgebra that is preserved by the boundary conditions is the fixed algebra under the action…