scholarly journals Mod-two cohomology rings of alternating groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Chad Giusti ◽  
Dev Sinha
Keyword(s):  
Individual Component ◽  
Symmetric Groups ◽  
Ring Structure ◽  
Steenrod Algebra ◽  
Transfer Product ◽  
Additive Structure ◽  
New Techniques ◽  
Alternating Groups ◽  
Cohomology Rings ◽  
Abelian Subgroups

AbstractWe calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.

On the generating graphs of symmetric groups

2018 ◽  
Vol 21 (4) ◽  
pp. 629-649 ◽  
Author(s):  
Fuat Erdem
Keyword(s):  
Symmetric Groups ◽  
Alternating Groups

AbstractLet {S_{n}} and {A_{n}} be the symmetric and alternating groups of degree n, respectively. Breuer, Guralnick, Lucchini, Maróti and Nagy proved that the generating graphs {\Gamma(S_{n})} and {\Gamma(A_{n})} are Hamiltonian for sufficiently large n. However, their proof provided no information as to how large n needs to be. We prove that the graphs {\Gamma(S_{n})} and {\Gamma(A_{n})} are Hamiltonian provided that {n\geq 107}.

scholarly journals The mod-2 cohomology rings of symmetric groups

2012 ◽  
Vol 5 (1) ◽  
pp. 169-198 ◽  
Author(s):  
Chad Giusti ◽  
Paolo Salvatore ◽  
Dev Sinha
Keyword(s):  
Symmetric Groups ◽  
Cohomology Rings

ON THE COMMUTING GRAPH ASSOCIATED WITH THE SYMMETRIC AND ALTERNATING GROUPS

2008 ◽  
Vol 07 (01) ◽  
pp. 129-146 ◽  
Author(s):  
A. IRANMANESH ◽  
A. JAFARZADEH
Keyword(s):  
Undirected Graph ◽  
Symmetric Groups ◽  
Alternating Groups ◽  
Commuting Graph ◽  
Central Elements

The commuting graph of a group G, denoted by Γ(G), is a simple undirected graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. The commuting graph of a subset of a group is defined similarly. In this paper we investigate the properties of the commuting graph of the symmetric and alternating and subsets of transpositions and involutions in the symmetric groups.

scholarly journals Generators for Alternating and Symmetric Groups

1971 ◽  
Vol 12 (1) ◽  
pp. 63-68 ◽  
Author(s):  
I. M. S. Dey ◽  
James Wiegold
Keyword(s):  
Free Product ◽  
Modular Group ◽  
Symmetric Groups ◽  
Alternating Groups ◽  
Generating Set ◽  
Alternating And Symmetric Groups

Let Γ denote the modular group, that is, the free product of a group of order 2 and a group of order 3. Morris Newman investigates in [2] the factor-groups of Γ and calls them Γ-groups for short; thus a group is a Γ-group if and only if it has a generating set consisting of an element of order dividing 2 and an element of order dividing 3. Newman's interest centres on finite simple Γ-groups. He proves that the linear fractional groups LF(2,p) for primes p are Γ -groups, and poses the problem of deciding which of the alternating groups enjoy this property.

Maximal Abelian Subgroups of the Symmetric Groups

1971 ◽  
Vol 23 (3) ◽  
pp. 426-438 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Symmetric Group ◽  
General Problem ◽  
Abelian Subgroup ◽  
Symmetric Groups ◽  
Specific Class ◽  
Maximal Abelian Subgroup ◽  
Number Of Generators ◽  
Abelian Subgroups ◽  
Almost All

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

More on the OD-Characterizability of a Finite Group

2011 ◽  
Vol 18 (04) ◽  
pp. 663-674 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
S. Rahbariyan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Alternating Groups ◽  
Degree Pattern ◽  
Prime Graphs ◽  
A Finite Group ◽  
P 53 ◽  
Number Of Authors

The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut (O7(3)) and Aut (S6(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

scholarly journals The group of endotrivial modules for the symmetric and alternating groups

2010 ◽  
Vol 53 (1) ◽  
pp. 83-95 ◽  
Author(s):  
Jon F. Carlson ◽  
David J. Hemmer ◽  
Nadia Mazza
Keyword(s):  
Modular Group ◽  
Symmetric Groups ◽  
Group Algebras ◽  
Torsion Subgroup ◽  
Alternating Groups ◽  
Free Part ◽  
Rank 2 ◽  
Torsion Free

AbstractWe complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n ≥ p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano.

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