Let G be a reductive group acting on an affine variety X, let x in X be a point whose G-orbit is not closed, and let S be a G-stable closed subvariety of X which meets the closure of the G-orbit of x but does not contain x. In this paper, we study G.R. Kempf's optimal class Omega_G(x,S) of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G.
Suppose K is a G-completely reducible subgroup of G which fixes x, and let H = C_G(K)^0. Our main result says that the H-orbit of x is also not closed, and the optimal class Omega_H(x,S) for H simply consists of the cocharacters in Omega_G(x,S) which evaluate in H. We apply this result in the case that G acts on its Lie algebra via the adjoint representation to obtain some new information about cocharacters associated with nilpotent elements in good characteristic.
Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H^G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We prove that a nilpotent-by-finite group with no proper contranormal subgroup is nilpotent. There are locally nilpotent groups with a proper contranormal subgroup. We study the structure of hypercentral groups with a finite proper contranormal subgroup.
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results e…
We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.
Let $H$ be an abelian subgroup of a finite group $G$ and $π$ the set of prime divisors of $|H|$. We prove that $|H O_π(G)/ O_π(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is nilpotent.
We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely $H_3(\mathbb{Z})$ the three-dimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann.
Let $G$ be a finite group and $p$ a fixed prime divisor of $|G|$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p'$-order, then (1) $G$ is solvable; (2) $G$ has a Sylow tower; (3) There exists at most one prime divisor $q$ of $|G|$ such that $G$ is neither $q$-nilpotent nor $q$-closed, where $q\neq p$.
We first give complete characterizations of the structure of finite group $G$ in which every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$. Furthermore, we prove that every self-centralizing subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order for a fixed prime divisor $p$ of $|G|$ if and only if every subgroup (or non-nilpotent subgroup, or non-abelian subgroup) of $G$ is a TI-subgroup or subnormal or has $p'$-order. Based on th…
Let $N$ be a finitely generated nilpotent group. The subgroup zeta function $ζ_N^{\leq}(s)$ and the normal zeta function $ζ_N^\lhd(s)$ of $N$ are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of $N$. We present results about their abscissae of convergence $α_N^\leq$ and $α_N^\lhd$, also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of $N$, respectively. We first prove some upper bounds for the functions $N\mapsto α_N^\leq$ and $N\mapstoα_N^\lhd$ when restricted to the class of torsion-free nilpotent group…
We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there…
We prove that the subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group are finite sums of Euler products of cone integrals over $\mathbb{Q}$ and we deduce from this that they have rational abscissa of convergence and some meromorphic continuation. We also define Mal'cev completions of a finitely generated virtually nilpotent group and we prove that the subgroup growth and the normal subgroup growth of the latter are invariants of its $\mathbb{Q}$-Mal'cev completion.
Let $G$ be a finite group and $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}σ_{i}$ and $σ_{i}\cap σ_{j}=\emptyset$ for all $i\neq j$. A chief factor $H/K$ of $G$ is said to be $σ$-central in $G$, if the semidirect product $(H/K)\rtimes(G/C_G(H/K))$ is a $σ_i$-group for some $i\in I$. The group $G$ is said to be $σ$-nilpotent if either $G=1$ or every chief factor of $G$ is $σ$-central. In this paper, we study the properties of a finite group $G=AB$, factorized by two $σ$-nilpotent subgroups $A$ and $B$, and also generalize some k…
We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is $2$-Engel, and $2$-connected if and only if the group is nilpotent of class $2$ or less. We determine the homotopy type of the coset poset for the group of $4\times 4$ upper unitriangular matrices over $\mathbb{F}_p$, and for the Burnside groups of exponent $3$.
Let $q$ be a prime and $A$ a finite $q$-group of exponent $q$ acting by automorphisms on a finite $q'$-group $G$. Assume that $A$ has order at least $q^3$. We show that if $γ_{\infty} (C_{G}(a))$ has order at most $m$ for any $a \in A^{\#}$, then the order of $γ_{\infty} (G)$ is bounded solely in terms of $m$. If the Fitting subgroup of $C_{G}(a)$ has index at most $m$ for any $a \in A^{\#}$, then the second Fitting subgroup of $G$ has index bounded solely in terms of $m$.
We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of $\mathcal{M}$. In addition, we prove that the order of $\mathcal{M}$ is equal to the order of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of $n$ into distinct positive integers. We show that these results contrast with those for the…
Let $π$ be a set of primes. We show that $π$-separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.
Let $q$ be a prime and $A$ an elementary abelian group of order at least $q^3$ acting by automorphisms on a finite $q'$-group $G$. It is proved that if $|γ_{\infty}(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $γ_{\infty}(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_G(a)$ for any $a \in A^{\#}$, then the index of $F_2(G)$ is $m$-bounded.
