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Attila Maróti

Publications and source records attributed to Attila Maróti.

32 records · Page 2Linked to original sources
arXiv2018

An improved diameter bound for finite simple groups of Lie type

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a non-abelian finite simple group. Let $G$ be a finite simple group of Lie type of Lie rank $n$ over the field $F_{q}$. Babai's conjecture has been verified in case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently, Biswas and Yang proved that $\mathrm{diam}(G)$ is bounded by $q^{O( n {(\log_{2}n + \log_{2}q)}^{3})}$. We show that in fact $\mathrm{diam}(G) < q^{O(n {(\log_{2}n)}^{2})}$ holds. Note that our bound is significantly smaller than the order of $G$ for $n$ large, even if $q$ is large. As an application, we show that more generally $\mathrm{diam}(H) < q^{O( n {(\log_{2}n)}^{2})}$ holds for any subgroup $H$ of $\mathrm{GL}(V)$, where $V$ is a vector space of dimension $n$ defined over the field $F_q$.

math.GR↗
arXiv2018

Finite groups with large Noether number are almost cyclic

Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$. Let $β(G)$ denote the largest indispensable degree in such generating sets. Cziszter and Domokos recently described finite groups $G$ with $|G|/β(G)$ at most $2$. We prove an asymptotic extension of their result. Namely, $|G|/β(G)$ is bounded for a finite group $G$ if and only if $G$ has a characteristic cyclic subgroup of bounded index. In the course of the proof we obtain the following surprising result. If $S$ is a finite simple group of Lie type or a sporadic group then we have $β(S) \leq {|S|}^{39/40}$. We ask a number of questions motivated by our results.

math.GR↗
arXiv2018

Set-Direct Factorizations of Groups

We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$ with $x\in X$ and $y\in Y$. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a group $G$ has such a factorization if and only if $G$ is a central product of $\left\langle X\right\rangle $ and $\left\langle Y\right\rangle $ and the central subgroup $\left\langle X\right\rangle \cap \left\langle Y\right\rangle $ satisfies certain abelian factorization conditions. We analyze some special cases and give examples. In particular, simple groups have no non-trivial set-direct factorization.

math.GR↗
arXiv2018

A proof of Pyber's base size conjecture

Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $\log |G| / \log n \leq b(G) < 45 (\log |G| / \log n) + c$. This finishes the proof of Pyber's base size conjecture. An ingredient of the proof is that for the distinguishing number $d(G)$ (in the sense of Albertson and Collins) of a transitive permutation group $G$ of degree $n > 1$ we have the estimates $\sqrt[n]{|G|} < d(G) \leq 48 \sqrt[n]{|G|}$.

math.GR↗
arXiv2017

Normalizers of Primitive Permutation Groups

Let $G$ be a transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G| < n$ if $G$ is primitive unless $n = 3^{4}$, $5^4$, $3^8$, $5^8$, or $3^{16}$. This bound is sharp when $n$ is prime. In fact, when $G$ is primitive, $|\mathrm{Out}(G)| < n$ unless $G$ is a member of a given infinite sequence of primitive groups and $n$ is different from the previously listed integers. Many other results of this flavor are established not only for permutation groups but also for linear groups and Galois groups.

math.GR↗
arXiv2015

Finite groups have more conjugacy classes

We prove that for every $ε> 0$ there exists a $δ> 0$ so that every group of order $n \geq 3$ has at least $δ\log_{2} n/{(\log_{2} \log_{2} n)}^{3+ε}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order $n$ has more than $\log_{3}n$ conjugacy classes. We answer Bertram's question in the affirmative for groups with a trivial solvable radical.

math.GR↗
arXiv2015

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter subgroups, we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

math.GR↗
arXiv2015

Groups equal to a product of three conjugate subgroups

Let $G$ be a finite non-solvable group. We prove that there exists a proper subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$, thus replacing an earlier upper bound of $36$ with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group $G$ with a $BN$-pair and a finite Weyl group $W$ satisfies $G=\left( Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest element of $W$. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of $B$ in $G$. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.

math.GR↗
arXiv2014

On the number of $p'$-degree characters in a finite group

Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for infinitely many groups $G$.

math.GR↗
arXiv2014

On a conjecture of Gluck

Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|\leq b(G)^{2}$. We confirm this conjecture in the case where $|F(G)|$ is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.

math.GR↗
arXiv2014

The minimal base size for a p-solvable linear group

Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $G\leq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on $V$ of size at most $2$ unless $q \leq 4$ in which case there exists a base of size at most $3$. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Pálfy and Wolf is also given.

math.GR↗