arXiv subjects
Publications and source records attributed to Aner Shalev.
We conjecture that if $G$ is a simple compact Lie group with trivial center, then every $d$-variable non-constant word map with coefficients in $G$ defines a non-constant function on $G^d$. We prove the conjecture for $A_r$, $B_r$, $E_6$, and $G_2$ using a ping-pong argument.
Let $A$ be a brace of cardinality $p^{n}$ where $p>n+1$ is prime, and let $ann (p^{2})$ be the set of elements of additive order at most $p^{2}$ in this brace. We construct a pre-Lie ring related to the brace $A/ann(p^{2})$. In the case of strongly nilpotent braces of nilpotency index $k<p$ the brace $A/ann(p^{2})$ can be recovered by applying the construction of the group of flows to the resulting pre-Lie ring. We don't know whether, when applied to braces which are not right nilpotent, our construction is related to the group of flows. We use powerful Lie rings associated with finite $p$-groups in the study of brace automorphisms with few fixed points. As an application we bound the number of elements which commute with a given element in a brace, as well as the number of elements which multiplied from left by a given element give zero. We also study various Lie rings associated to powerful groups and braces whose adjoint groups are powerful, and show that the obtained Lie and pre-Lie rings are also powerful. We also show that braces whose adjoint groups are powerful and powerful left nilpotent pre-Lie rings are in one-to-one correspondence and that they are left and right nilpotent under some cardinality assumptions.
The deep theory of approximate subgroups establishes 3-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper we obtain 2-step growth results for representations of such groups $G$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $G$ be a finite simple group of Lie type and $\chi$ a character of $G$. Let $|\chi|$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $G$ which are constituents of $\chi$. We show that for all $\delta>0$ there exists $\epsilon>0$, independent of $G$, such that if $\chi$ is an irreducible character of $G$ satisfying $|\chi| \le |G|^{1-\delta}$, then $|\chi^2| \ge |\chi|^{1+\epsilon}$. We also obtain results for reducible characters, and establish faster growth in the case where $|\chi| \le |G|^{\delta}$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $G$ occurs as a constituent of certain products of characters. For example, we prove that if $|\chi_1| \cdots |\chi_m|$ is a high enough power of $|G|$, then every irreducible character of $G$ appears in $\chi_1\cdots\chi_m$. Finally, we obtain growth results for compact semisimple Lie groups.
We show that, if $w_1, \ldots , w_6$ are words which are not an identity of any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$ for all (non-abelian) finite simple groups $G$. In particular, for every word $w$, either $w(G)^6 = G$ for all finite simple groups, or $w(G)=1$ for some finite simple groups. These theorems follow from more general results we obtain on characteristic collections of finite groups and their covering numbers, which are of independent interest and have additional applications.
Fix a positive integer $d$ and let $\Gamma_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $\Gamma_d$ were studied by Babai, Cameron and P\'{a}lfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function $f$ such that the base size of any finite primitive group with point stabilizers in $\Gamma_d$ is at most $f(d)$. This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of $c$ points lie in $\Gamma_d$. We also show that if $G$ is any permutation group of degree $n$ whose $c$-point stabilizers lie in $\Gamma_d$, then $|G| \leqslant ((1+o_c(1))d/e)^{n-1}$. This asymptotically extends and improves a $d^{n-1}$ upper bound on $|G|$ obtained by Babai, Cameron and P\'{a}lfy assuming $G \in \Gamma_d$.
There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, and finite simple groups in particular. In this paper we study similar notions for finite and profinite associative algebras. Let $k=F_q$ be a finite field. Let $A$ be a finite dimensional, associative, unital algebra over $k$. Let $P(A)$ be the probability that two elements of $A$ chosen (uniformly and independently) at random will generate $A$ as a unital $k$-algebra. It is known that, if $A$ is simple, then $P(A) \to 1$ as $|A| \to \infty$. We extend this result to a large class of finite associative algebras. For $A$ simple, we find the optimal lower bound for $P(A)$ and we estimate the growth rate of $P(A)$ in terms of the minimal index $m(A)$ of any proper subalgebra of $A$. We also study the random generation of simple algebras $A$ by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let $A$ be a profinite algebra over $k$. We show that $A$ is positively finitely generated if and only if $A$ has polynomial maximal subalgebra growth. Related quantitative results are also established.
Let $A$ be a residually finite dimensional algebra (not necessarily associative) over a field $k$. Suppose first that $k$ is algebraically closed. We show that if $A$ satisfies a homogeneous almost identity $Q$, then $A$ has an ideal of finite codimension satisfying the identity $Q$. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra $L$ over $k$ is almost $d$-Engel, then $L$ has a nilpotent (resp. locally nilpotent) ideal of finite codimension if char $k=0$ (resp. char $k > 0$). Next, suppose that $k$ is finite (so $A$ is residually finite). We prove that, if $A$ satisfies a homogeneous probabilistic identity $Q$, then $Q$ is a coset identity of $A$. Moreover, if $Q$ is multilinear, then $Q$ is an identity of some finite index ideal of $A$. Along the way we show that, if $Q\in k\langle x_1,\ldots,x_n\rangle$ has degree $d$, and $A$ is a finite $k$-algebra such that the probability that $Q(a_1, \ldots , a_n)=0$ (where $a_i \in A$ are randomly chosen) is at least $1-2^{-d}$, then $Q$ is an identity of $A$. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon.
Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an associative, not necessarily unital, algebra over $k$. An unrefinable chain of $A$ is a chain of subalgebras $A=A_0>A_1>...>A_t=0$ for some integer $t$ where each $A_i$ is a maximal subalgebra of $A_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $A$. It turns out that finite length, finite depth and finite dimension are equivalent properties for $A$. For $A$ finite dimensional, we give a formula for the length of $A$, we bound the depth of $A$, and we study when the length of $A$ equals its dimension and its depth respectively. Finally, we investigate under what circumstances the dimension of $A$ is bounded above by a function of its length, or its depth, or its length minus its depth.
In recent years there has been significant progress in the study of products of subsets of finite groups and of finite simple groups in particular. In this paper we consider which families of finite simple groups $G$ have the property that for each $\epsilon > 0$ there exists $N > 0$ such that, if $|G| \ge N$ and $S, T$ are normal subsets of $G$ with at least $\epsilon |G|$ elements each, then every non-trivial element of $G$ is the product of an element of $S$ and an element of $T$. We show that this holds in a strong sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{PSL}}_n(q)$ where $q$ is fixed and $n$ tends to infinity. Our second main result is that any element in a transitive permutation representation of a sufficiently large finite simple group is a product of two derangements.
Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph ${\mathcal M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of $\alpha\chi_1$. We study the diameters of McKay graphs for simple groups $G$. For $G$ a group of Lie type, we show that for any $\alpha$, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for $G={\rm PSL}_n(q)$ or ${\rm PSU}_n(q)$. We also bound the diameter for symmetric and alternating groups.
We define the notion of an almost polynomial identity of an associative algebra $R$, and show that its existence implies the existence of an actual polynomial identity of $R$. A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras.
We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map $w_G:G^d\rightarrow G$ has a fiber of size at least $\rho|G|^d$ for some fixed $\rho>0$. We show that, for certain words $w$, this implies that $G$ has a normal solvable subgroup of index bounded above in terms of $w$ and $\rho$. We also show that, for a larger family of words $w$, this implies that the nonsolvable length of $G$ is bounded above in terms of $w$ and $\rho$, thus providing evidence in favor of a conjecture of Larsen. Along the way we obtain results of some independent interest, showing roughly that most elements of large finite permutation groups have large support.
We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.
Let $G$ be a simple algebraic group over the algebraic closure of $GF(p)$ ($p$ prime), and let $G(q)$ denote a corresponding finite group of Lie type over $GF(q)$, where $q$ is a power of $p$. Let $X$ be an irreducible subvariety of $G^r$ for some $r\ge 2$. We prove a zero-one law for the probability that $G(q)$ is generated by a random $r$-tuple in $X(q) = X\cap G(q)^r$: the limit of this probability as $q$ increases (through values of $q$ for which $X$ is stable under the Frobenius morphism defining $G(q)$) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs $G(q)$ to be generated by an $r$-tuple in $X(q)$ for two sufficiently large values of $q$. We also prove a version of this result where the underlying characteristic is allowed to vary. In our main application, we apply these results to the case where $r=2$ and the irreducible subvariety $X = C\times D$, a product of two conjugacy classes of elements of finite order in $G$. This leads to new results on random $(2,3)$-generation of finite simple groups $G(q)$ of exceptional Lie type: provided $G(q)$ is not a Suzuki group, we show that the probability that a random involution and a random element of order 3 generate $G(q)$ tends to $1$ as $q \rightarrow \infty$. Combining this with previous results for classical groups, this shows that finite simple groups (apart from Suzuki groups and $PSp_4(q)$) are randomly $(2,3)$-generated. Our tools include algebraic geometry, representation theory of algebraic groups, and character theory of finite groups of Lie type.
The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distribution on finite simple groups with respect to the $L^1$ norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distribution on finite simple groups of Lie type of bounded rank, and study related random walks. Our second main result concerns the probabilistic $L^{\infty}$ Waring problem for finite simple groups. We show that for every $l \ge 1$ there exists $N = N(l)$, such that if $w_1, \ldots , w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^{\infty}$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ as above, the word map induced by $w$ on a semisimple algebraic group over an arbitrary field is a flat morphism. Applications to representation varieties, subgroup growth, and random generation are also presented.
We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.
Let $G$ be a connected Lie group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. In this paper, we introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups $G$. We obtain best possible bounds on the length of $G$ in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on $\dim G'$ in terms of the chain difference of $G$, which is its length minus its depth.
An unrefinable chain of a finite group $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal subgroup of $G_{i-1}$. The length (respectively, depth) of $G$ is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth $3$. Here we go much further by determining the finite groups of depth $3$ and $4$. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most $9$, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a number-theoretic result of Heath-Brown, which implies that there are infinitely many non-abelian simple groups of length at most $9$. Finally we study the chain difference of $G$ (namely the length minus the depth). We obtain results on groups with chain difference $1$ and $2$, including a complete classification of the simple groups with chain difference $2$, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of $G/R(G)$ in terms of the chain difference of $G$, where $R(G)$ is the soluble radical of $G$.