A group $G$ is said to be factorized into subsets $A_1, A_2, \ldots, A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. We consider the following conjecture: for every finite group $G$ and every factorization $n=ab$ of its order, there is a factorization $G=AB$ with $|A|=a$ and $|B|=b$. We show that a minimal counterexample to this conjecture must be a nonabelian simple group and prove the conjecture for every finite group the nonabelian composition factors of which have orders less than $10\,000$.
Let $G $ be a group of cardinality $κ>\aleph_0 $ endowed with a topology $τ$ such that $|U|=κ$ for every non-empty $U\inτ$ and $τ$ has a base of cardinality $κ$. We prove that $G$ could be factorized $G=AB$ (i.e. each $g\in G$ has unique representation $g=ab$, $a\in A$, $b\in B$) into dense subsets $A,B$, $|A|=|B|=κ$. We do not know if this statement holds for $κ= \aleph_0$ even if $G$ is a topological group.
For some very wide classes $\mathfrak{D}$ and $\mathfrak{B}\subset\mathfrak{D}$ of groups, the author proves that an arbitrary (nonabelian) group $G\in \mathfrak{D}$ (respectively $G\in \mathfrak{B}$) satisfies the minimal condition for (nonabelian) subgroups iff it is Chernikov
A finite group G is said to be a POS-group if the number of elements of every order occurring in G divides |G|. We answer two questions by Finch and Jones by providing an infinite family of nonabelian POS-groups with orders not divisible by 3.
Let $G$ be a nonabelian group, $A\subseteq G$ an abelian subgroup and $n\geqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|>1$ for each $i=1, 2, \ldots, n$. We first classify all nonabelian groups, up to isomorphism, which have an $n$-abelian partition for $n=2, 3$. Then, we provide some formulas concerning the number of spanning trees of commuting graphs associated with certain finite groups. Finally, we point out some ways to f…
Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such that $|B_i| > 1$ for each $i$ and within each set $B_i$, no two distinct elements commute. We show that every finite nonabelian group has a strict $n$-split decomposition for some $n$. We classify all finite groups $G$, up to isomorphism, which have a strict $n$-split decomposition for $n = 1, 2, 3$. Finally, we show that for a nonabelian group $G$ having a st…
In analogy with epsilon-biased sets over Z_2^n, we construct explicit epsilon-biased sets over nonabelian finite groups G. That is, we find sets S subset G such that | Exp_{x in S} rho(x)| <= epsilon for any nontrivial irreducible representation rho. Equivalently, such sets make G's Cayley graph an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem shows that random sets of size O(log |G| / epsilon^2) suffice. For groups of the form G = G_1 x ... x G_n, our construction has size poly(max_i |G_i|, n, epsilon^{-1}), and we show that a set S \subset G^n considered by Meka and…
Subsets of abelian groups closed under addition or subtraction
Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University（维尔纽斯大学数学与信息学院数学研究所）；School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）
The spectrum of a periodic group $G$ is the set $ω(G)$ of its element orders. Consider a group $G$ such that $ω(G)=ω(A_7)$. Assume that $G$ has a subgroup $H$ isomorphic to $A_4$, whose involutions are squares of elements of order $4$. We prove that either $O_2(H) \subseteq O_2(G)$ or $G$ has a finite nonabelian simple subgroup.
Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper investigates problems relating to group with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian part…
Suppose $G$ is a connected noncompact locally compact group, $A,B$ are nonempty and compact subsets of $G$, $μ$ is a left Haar measure on $G$. Assuming that $G$ is unimodular, and $ μ(A^2) < K μ(A) $ with $K>1$ a fixed constant, our first result shows that there is a continuous surjective group homomorphism $χ: G\to L$ with compact kernel, where $L$ is a Lie group with $$\dim(L) \leq \lfloor\log K\rfloor(\lfloor\log K\rfloor+1)/2.$$ We also demonstrate that this dimension bound is sharp, establish the relationship between $A$ and its image under the quotient map, and obtain a more general vers…
We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.
A language-theoretic approach to study the density of subsets in free groups
Centro de Investigação em Matemática e Aplicações (CIMA)（数学及其应用研究中心）；Departamento de Matemática, Escola de Ciências e Tecnologia da Universidade de Évora（埃武拉大学科学与技术学院数学系）
Let $π$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in the $π$-radical $\mathrm{O}_π(G)$ of $G$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to sporadic or alternating groups.
A matching in a group G is a bijection f from a subset A to a subset B in G such that af(a) does not belong to A for all a in A. The group G is said to have the matching property if, for any finite subsets A,B in G of same cardinality with B avoiding 1, there is a matching from A to B. Using tools from additive number theory, Losonczy proved a few years ago that the only abelian groups satisfying the matching property are the torsion-free ones and those of prime order. He also proved that, in an abelian group, any finite subset A avoiding 1 admits a matching from A to A. In this paper, we show…
Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $ε>0$, describe the structure of $A$ and behave regularly with respect to translates of $A$. For the subclass of groups with uniformly fixed finite exponent $r$, these algebraic objects are normal subgroups with index bounded in terms of $k$, $r$, and $ε$. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model theoret…
We study the distribution of products of conjugacy classes in finite simple groups, obtaining various effective uniformity results, which give rise to an approximation to a conjecture of Thompson. Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL$(2,q)$ to all (nonabelian) finite simple groups. In particular it follows that, if $G$ is a finite simple group, and $A, B \subseteq G^t$ for $t \ge 2$ are subsets of fixed positive densities,…
In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $ω$ of matrix multiplication. We present two new characterizations of the TPP, which are useful for theoretical considerations and for TPP test algorithms. With this we describe all known TPP tests and implement them in GAP algorithms. We also compare their runtime. Furthermore we show that the search for subgroup TPP triples of nontrivial size in a nonabelian group can be res…
In this paper, we consider a natural generalization of the concept of order of an element in a group: an element $g \in G$ is said to have order $k$ in a subgroup $H$ of $G$ (\resp \wrt a coset $Hu$) if $k$ is the first strictly positive integer such that $g^k \in H$ (\resp $g^k \in Hu$). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set o…
Answering a question of Gowers, Tao proved that any $A\times B\times C\subset SL_d(\mathbb{F}_q)^3$ contains $|A||B||C|/|SL_d(\mathbb{F}_q)|+O_d(|SL_d(\mathbb{F}_q)|^2/q^{\min(d-1,2)/8})$ three-term progressions $(x,xy,xy^2)$. Using a modification of Tao's argument, we prove such a mixing result for three-term progressions in all nonabelian finite simple groups except for $PSL_2(\mathbb{F}_q)$ with an error term that depends on the degree of quasirandomness of the group. This argument also gives an alternative proof of Tao's result when $d>2$, but with the error term $O(|SL_d(\mathbb{F}_q)|^2/…
In noncommutative geometry a `Lie algebra' or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset C stable under inversion. We study the associated Killing form. For the universal calculus associated to C=G \ {e} we show that the magnitude of the Killing form μ=\sum_{a,b\in C}K^{-1}_{a,b} is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ is not equal to 1/(N-1), where N is the number of conjugacy classes…
