Related papers: Subgroup families controlling p-local finite group…

Subgroups $H_1$ and $H_2$ of a group $G$ are said to be locally conjugate if there is a bijection $f: H_1 \rightarrow H_2$ such that $h$ and $f(h)$ are conjugate in $G$ for every $h \in H_1$. This paper studies local conjugacy among…

When $(S,\mathcal{F},\mathcal{L})$ is a $p$-local finite group and $(T,\mathcal{E},\mathcal{\L}_0)$ is weakly normal in $(S,\mathcal{F},\mathcal{L})$ we show that a definition of $C_S(\mathcal{E})$ given by Aschbacher has a simple…

Given a type A in homotopy type theory (HoTT), we can define the free infinity-group on A as the loop space of the suspension of A+1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit…

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

We study saturated fusion systems on $p$-groups having sectional rank $3$ for all odd primes $p$. For $p\geq 5$, we obtain a complete classification of the ones that do not have any non-trivial normal $p$-subgroups.

The central problem in computational algebraic topology is the computation of the homotopy groups of a given space, represented as a simplicial set. Algorithms have been found which achieve this, but the running times depend on the size of…

For a subgroup $S$ of a group $G$, let $I_G(S)$ denote the set of commutators $[g,s]=g^{-1}g^s$, where $g\in G$ and $s\in S$, so that $[G,S]$ is the subgroup generated by $I_G(S)$. We prove that if $G$ is a $p$-soluble finite group with a…

Let $G$ be a closed highly homogeneous subgroup of $S_{\infty}$ not involving circular orderings. We show that the closure of a conjugacy class from $G$ contains a conjugacy class which is comeagre in it. Furthermore, we show that the…

We study the topological fundamental groups of the locally finite infinite ordered configuration space \(Conf^{lf}_\infty(\C)\) in the plane and the homotopy quotient of $Conf^{lf}_\infty$ by the canonical action of the infinite permutation…

We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at…

A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H \cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$. A new…

A B-group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite B-groups is closed under taking quotients and that every finite B-group is solvable. Via a…

We review the definition of hypergroups by Sunder, and we associate a hypergroup to a type III subfactor $N\subset M$ of finite index, whose canonical endomorphism $\gamma\in\mathrm{End}(M)$ is multiplicity-free. It is realized by positive…

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…

Coclass theory can be used to define infinite families of finite p-groups of a fixed coclass. It is conjectured that the groups in one of these infinite families all have isomorphic mod-p cohomology rings. Here we prove that almost all…

We prove the conjugacy of Sylow $2$-subgroups in pseudofinite $\mathfrak{M}_c$ (in particular linear) groups under the assumption that there is at least one finite Sylow $2$-subgroup. We observe the importance of the pseudofiniteness…

For any infinite-type surface $S$, a natural question is whether the homology of its mapping class group contains any non-trivial classes that are supported on (i) a compact subsurface or (ii) a finite-type subsurface. Our purpose here is…

We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each…

The well-known Landau's theorem states that, for any positive integer $k$, there are finitely many isomorphism classes of finite groups with exactly $k$ (conjugacy) classes. We study variations of this theorem for $p$-regular classes as…

We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus $g$ handlebody group into the genus $g$ mapping class group is conjugation…