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Fixed‐point ratios, Sylow numbers, and coverings of p$p$‐elements in finite groups

Authors

Robert M. Guralnick, Attila Maróti, Juan José Prieto Martínez, Alexander Moretó, Noelia Rizo
Type: Article
Publication Date: 2025-05-01
Citations: 0
DOI: https://doi.org/10.1112/jlms.70167

Abstract

Abstract Fixed‐point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime and a finite group , we use fixed‐point ratios to study the number of Sylow ‐subgroups of and the minimal size of a covering by proper subgroups of the set of ‐elements of .

Locations

  • Journal of the London Mathematical Society
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Let G be a group, and associate with G a graph 1 γ = Γ ( G ) as follows: the vertices of Γ are the elements of G , … Let G be a group, and associate with G a graph 1 γ = Γ ( G ) as follows: the vertices of Γ are the elements of G , and two vertices g, h of Γ are joined by an undirected edge if, and only if, g and h do not commute as elements of G , that is [g, h] ≠ 1 [where [g, h] is the commutator g −1 h −1 gh , and 1 is the unit element of the groups that occur as well as the integer, according to context].We are interested in complete subgraphs of Γ, or equivalently in sets of elements of G no two of which commute.
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Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ … Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees $n\leq 4$, due to work of van der Waerden and Chow and Dietmann. The purpose of this paper is to prove van der Waerden's Conjecture for all degrees $n$.

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Finite Group Theory

2008-08-06
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2022-11-18
Timothy C. Burness , Robert M. Guralnick
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Chat View PDF

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2023-05-26
Timothy C. Burness , Robert M. Guralnick , Alexander Moretó +1 more
Let G be a finite group, let p be a prime and let Pr p (G) be the probability that two random p-elements of G commute.In this paper we prove … Let G be a finite group, let p be a prime and let Pr p (G) be the probability that two random p-elements of G commute.In this paper we prove that Pr p (G) > ( p 2 + p -1)/ p 3 if and only if G has a normal and abelian Sylow p-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group.This bound is best possible in the sense that for each prime p there are groups with Pr p (G) = ( p 2 + p -1)/ p 3 and we classify all such groups.Our proof is based on bounding the proportion of p-elements in G that commute with a fixed p-element in G \ O p (G), which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.
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Finite groups can be generated by a π-subgroup and a π′-subgroup

2023-11-01
Thomas Breuer , Robert M. Guralnick

On the commuting probability of π$\pi$‐elements in finite groups

2024-02-15
Juan José Prieto Martínez
Abstract Let be a finite group, be a set of primes, and be the smallest prime in . In this work, we prove that possesses a normal and abelian Hall … Abstract Let be a finite group, be a set of primes, and be the smallest prime in . In this work, we prove that possesses a normal and abelian Hall ‐subgroup if and only if the probability that two random ‐elements of commute is larger than . We also prove that if is a ‐element not lying in , then the proportion of ‐elements commuting with is at most .
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Covering the set of p-elements in finite groups by proper subgroups

2024-09-20
Attila Maróti , Juan José Prieto Martínez , Alexander Moretó

None

2000-01-01
Daniel Frohardt , Kay Magaard