Related papers: An OD-Characterizable Class of Simple Groups

We prove that for any prime number $p$, every finite non-abelian $p$-group $G$ of class 2 has a noninner automorphism of order $p$ leaving either the Frattini subgroup $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise fixed.

In 2000, L. H\'{e}thelyi and B. K\"{u}lshammer proved that if $p$ is a prime number dividing the order of a finite solvable group $G$, then $G$ has at least $2\sqrt{p-1}$ conjugacy classes. In this paper we show that if $p$ is large, the…

A finite group of order divisible by 3 in which centralizers of 3-elements are 3-subgroups will be called a C{\theta}{\theta}-group. The prime graph (or Gruenberg-Kegel graph) of a finite group G is denoted by {\Gamma}(G) (or GK(G)) and its…

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative…

A derived version of Maschke's theorem for finite groups is proved: the derived categories, bounded or unbounded, of all blocks of the group algebra of a finite group are simple, in the sense that they admit no nontrivial recollements. This…

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let $\cd$ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If…

Pro-$p$ groups of finite powerful class are studied. We prove that these are $p$-adic analytic, and further describe their structure when their powerful class is small. It is also shown that there are only finitely many finite $p$-groups of…

A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We…

We prove an algebraic version of a classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove…

Let $G$ be a finite group and let $cd(G)$ be the set of all irreducible complex character degrees of $G$. It was conjectured by Huppert in Illinois J. Math. 44 (2000) that, for every non-abelian finite simple group $H$, if $cd(G)=cd(H)$…

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly $k$ real conjugacy…

Let $V$ be a finite-dimensional vector space over the complex numbers and let $G\leq \operatorname{SL}(V)$ be a finite group. We describe the class group of a minimal model (that is, $\mathbb Q$-factorial terminalization) of the linear…

We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite…

We classify closed abelian subgroups of the simple groups $G_2$, $F_4$, $Aut(so(8))$ having centralizer the same dimension as the dimension of the subgroup, as well as finite abelian subgroups of certain spin and half-spin groups having…

Let $G$ be a finite $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. In this article we classify non-abelian $p$-groups $G$ of…

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many…

In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. In particular, these simple groups have infinite $w$-verbal width for all…

A systematic analysis is presented of compactifications of the IIB superstring on $AdS_5 \times S^5/\Gamma$ where $\Gamma$ is a non-abelian discrete group. Every possible $\Gamma$ with order $g \leq 31$ is considered. There exist 45 such…