1,720,976 research outputs found
Covering monolithic groups with proper subgroups
No full textGiven a finite non-cyclic group G, call σ(G) the smallest number of proper subgroups of G needed to cover G. Lucchini and Detomi conjectured that if a nonabelian group G is such that σ(G) < σ(G/N) for every non-trivial normal subgroup N of G then G is monolithic, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.Given a finite non-cyclic group G, call σ(G) the smallest number of proper subgroups of G needed to cover G. Lucchini and Detomi conjectured that if a nonabelian group G is such that σ(G) < σ(G/N) for every non-trivial normal subgroup N of G then G is monolithic, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups. © 2013 University of Isfahan
Conjugate factorizations of finite groups
No full textIn this paper we illustrate recent results about factorizations of finite groups into conjugate
subgroups. The illustrated results are joint works with John Cannon, Dan Levy, Attila Mar ́oti and
Iulian I. Simion.In this paper we illustrate recent results about factorizations of finite groups into conjugate subgroups. The illustrated results are joint works with John Cannon, Dan Levy, Attila Maróti and Iulian I. Simion
Maximal irredundant families of minimal size in the alternating group
No full textLet G be a finite group. A family M of maximal subgroups of G is called irredundant if its intersection is not equal to the intersection of any proper subfamily. M is called maximal irredundant if M is irredundant and it is not properly contained in any other irredundant family. We denote by (G) when G is the alternating group on n letters
On the number of conjugacy classes of a permutation group
No full textWe prove that any permutation group of degree n at least 4 has at most 5^((n-1)/3) conjugacy classe
Inequalities detecting structural properties of a finite group
No full textWe study inequalities involving the element orders of a finite group and how they influence its structure.We prove several results detecting cyclicity or nilpotency of a finite group G in terms of inequalities involving the orders of the elements of G and the orders of the elements of the cyclic group of order |G|. We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group, and the product of the orders of the elements is maximal for the cyclic group
Direct products of finite groups as unions of proper subgroups
No full textWe determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality. © 2010 Springer Basel AG
On the primary coverings of finite solvable and symmetric groups
Full text linkA primary covering of a finite group is a family of proper subgroups of whose union contains the set of elements of having order a prime power. We denote by s 0 (G) sigma {0}(G) the smallest size of a primary covering of and call it the primary covering number of We study this number and compare it with its analogue s (G) sigma(G), the covering number, for the classes of groups that are solvable and symmetric.A primary covering of a finite group is a family of proper subgroups of whose union contains the set of elements of having order a prime power. We denote by s 0 (G) \sigma {0}(G) the smallest size of a primary covering of and call it the primary covering number of We study this number and compare it with its analogue s (G) \sigma(G), the covering number, for the classes of groups that are solvable and symmetric
Factorizing a finite group into conjugates of a subgroup
No full textWe prove that any finite nonsolvable group is a product of at most 36 conjugates of a proper subgroup and we give an upper bound in the case of solvable groups.For every non-nilpotent finite group G, there exists at least one proper subgroup M such that G is the setwise product of a finite number of conjugates of M. We define γcp(G) to be the smallest number k such that G is a product, in some order, of k pairwise conjugated proper subgroups of G. We prove that if G is non-solvable then γcp(G)≤36 while if G is solvable then γcp(G) can attain any integer value bigger than 2, while, on the other hand, γcp(G)≤4log2|G|
Finite groups with six or seven automorphism orbits
No full textThe finite nonsolvable groups with at most 6 automorphism orbits are classified, there are only a finite number of them up to isomorphism. It is also proved that there are infinitely many finite non solvable groups with 7 automorphism orbits.Let G be a group. The orbits of the natural action of Aut(G) on G are called "automorphism orbits" of G, and the number of automorphism orbits of G is denoted by ω(G). In this paper the finite non-solvable groups G with ω(G) ≤, 6 are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite non-solvable groups G with ω(G) = 7. Moreover, it is proved that for a given number n there are only finitely many finite groups G without non-trivial abelian normal subgroups and such that ω(G) ≤, n, generalizing a result of Kohl
On the maximal number of elements pairwise generating the symmetric group of even degree
No full textLet G be the symmetric group of degree n. Let ω(G) be the maximal size of a subset S of G such that 〈x,y〉=G whenever x,y∈S and x≠y and let σ(G) be the minimal size of a family of proper subgroups of G whose union is G. We prove that both functions σ(G) and ω(G) are asymptotically equal to [Formula presented] when n is even. This, together with a result of S. Blackburn, implies that σ(G)/ω(G) tends to 1 as n→∞. Moreover, we give a lower bound of n/5 on ω(G) which is independent of the classification of finite simple groups. We also calculate, for large enough n, the clique number of the graph defined as follows: the vertices are the elements of G and two vertices x,y are connected by an edge if 〈x,y〉≥An.Let G be the symmetric group of degree n. Let omega(G) be the maximal size of a subset S of G such that (x, y) = G whenever x, y E S and x not equal y and let sigma(G) be the minimal size of a family of proper subgroups of G whose union is G. We prove that both functions sigma(G) and omega(G) are asymptotically equal to 1/2(n/n/2) when n is even. This, together with a 2 n/2 result of S. Blackburn, implies that sigma(G)/omega(G) tends to 1 as n ->infinity. Moreover, we give a lower bound of n/5 on omega(G) which is independent of the classification of finite simple groups. We also calculate, for large enough n, the clique number of the graph defined as follows: the vertices are the elements of G and two vertices x, y are connected by an edge if (x, y) >= A(n). (C) 2021 Elsevier B.V. All rights reserved
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