Related papers: Fusion systems on metacyclic 2-groups

Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and complement $H$, acts by automorphisms on a finite group $G$, in such a way that $C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is nilpotent and its…

In this paper, we present an explicit structure for the Baer invariant of a free nilpotent group (the $n$-th nilpotent product of the infinite cyclic group, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*... \st{n}*\textbf{Z}$) with respect to the…

Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we…

We say that a fusion system is the composition product of two subsystems if every morphism can be factored as a morphism in one fusion system followed by a morphism in the other. We establish a relationship between the characteristic…

In this paper subcentral (resp., central) idempotent series and composition subcentral (resp., central) idempotent series in an inverse semigroup are introduced and investigated. It is shown that if $S=EG$ is a factorizable inverse monoids…

A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric…

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…

For any block of a finite group over an algebraically closed field of characteristic $2$ which has dihedral, semidihedral, or generalized quaternion defect groups, we determine explicitly the decomposition of the associated diagonal…

Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is called a weakly $S\Phi$-supplemented subgroup of $G$, if there exists a subgroup $T$ of $G$ such that $G =HT$ and $H \cap T \leq \Phi (H) H_{sG}$, where $H_{sG}$ denotes…

If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there…

A finitely generated group $G$ is called strongly scale-invariant if there exists an injective homomorphism $f:G\to G$ such that $f(G)$ is a finite index subgroup of $G$ and such that $\cap_{n\geq 0} f^n(G)$ is finite. Nekrashevych and Pete…

Let $A$ be a non-metacyclic finite group. Suppose that $A$ acts coprimely on a finite group $G$ in such a manner that $C_G(a)$ is nilpotent for any $a\in A^{\#}$. In the present paper we investigate some conditions on $A$ which imply that…

A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group…

In this short note we prove that a saturated fusion system admitting some special type of automorphism is nilpotent. This generalizes classical results by J.G. Thompson.

We begin an investigation of supersymmetric theories based on exceptional groups. The flat directions are most easily parameterized using their correspondence with gauge invariant polynomials. Symmetries and holomorphy tightly constrain the…

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the metabelian small class case. The approach is also used to…

We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…

Let $FH$ be a supersolvable Frobenius group with kernel $F$ and complement $H$. Suppose that a finite group $G$ admits $FH$ as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_{G}(H)$ is nilpotent of class $c$. We show that…

We prove, for certain pairs G,G of finite groups of Lie type, that the p-fusion systems for G and G' are equivalent. In other words, there is an isomorphism between a Sylow p-subgroup of G and one of G' which preserves p-fusion. This…

We show that if $\mathcal{C}$ is a fusion $2$-category in which the endomorphism category of the unit object is $\rm{Vec}$ or $\rm{SVec}$, then the indecomposable objects of $\mathcal{C}$ form a finite group.