Permutation statistics on the alternating group

Let A_nS_n denote the alternating and the symmetric groups on 1,…,n. MacMahon's theorem [P.A. MacMahon, Combinatory Analysis I–II, Cambridge Univ. Press, 1916], about the equi-distribution of the length and the major indices in S_n, has received far reaching refinements and generalizations, by Foata [Proc. Amer. Math. Soc. 19 (1968) 236], Carlitz [Trans. Amer. Math. Soc. 76 (1954) 332; Amer. Math. Monthly 82 (1975) 51], Foata-Schützenberger [Math. Nachr. 83 (1978) 143], Garsia–Gessel [Adv. Math. 31 (1979) 288] and followers. Our main goal is to find analogous statistics and identities for the alternating group A_n. A new statistics for S_n, the delent number, is introduced. This new statistics is involved with new S_n identities, refining some of the results in [D. Foata, M.P. Schützenberger, Math. Nachr. 83 (1978) 143; A.M. Garsia, I. Gessel, Adv. Math. 31 (1979) 288]. By a certain covering map , such S_n identities are ‘lifted’ to A_n+1, yielding the corresponding A_n+1 equi-distribution identities.     

A characterization of alternating groups by the set of orders of maximal abelian subgroups

We prove that alternating groups with three prime graph components are uniquely determined by the set of orders of maximal abelian subgroups.     

Equivalence and equation solvability problems for the alternating group A4

It is observed in this paper that the complexities of the equivalence and the equation solvability problems are not determined by the clone of the algebra. In particular, we prove that for the alternating group on four elements these problems have complexity in P; if we extend the group by the commutator as an extra operation, then the equivalence problem is coNP-complete and the equation solvability problem is NP-complete.     

Linear maps leaving the alternating group invariant

Let A_n be the group of n×n even permutation matrices, and let V_n be the real linear space spanned by A_n. The purpose of this note is to characterize those linear operators φ on V_n satisfying φ(A_n)=A_n. This answers a question raised by C.K. Li, B.S. Tam, N.K. Tsing [Linear Algebra Appl., to appear].     

Generating the alternating group by cyclic triples

It is shown that a collection of circular permutations of length three on an n-set generates the alternating group A_n if and only if the associated graph is connected. It follows that [$\text{1}\text{2}\text{n}$] circular permutations of length three may generateA_n.     

Cones of alternating and cut submodular set functions

We describe facets of the cones of alternating set functions and cut submodular set functions generated by directed and undirected graphs and by uniform even hypergraphs. This answers a question asked by L. Lovász at the Bonn Mathematical Programming Conference in 1982. We show that there is a network flow algorithm for minimizing a hypergraph cut set function.     

Reflection classes whose cubes cover the alternating group

A reflection class (REC) over a finite set A is a conjugacy class of a reflection (permutation of order ? 2) of A. It was known that for no REC X, X² = Alt(n) holds, and that for some RECs X, X⁴ = Alt(n) holds (n ? 5). Let i > 0, and let c(θ) denote the number of cycles of θ?S(n). Let X_i = {ψ ∈ S(n): ψ² = 1, ψ has exactly i fixed points}. We prove that θ?X_i³ if and only if: (1) i ≡ n (mod 2); (2) The parity of X_i equals the parity of θ; and (3) $\text{i ?}\text{1}\text{3}\text{(n + 2 c(θ))}$. As a consequence, {X: X is a REC, X³ = Alt(n)} and {X: X is a REC, X³ = S(n) ? Alt(n)} are determined.     

Groups isospectral to the degree 10 alternating group

The spectrum of a finite group is the set of its element orders. We describe the composition structure of every finite group with the same spectrum as that of the alternating group of degree 10 and not isomorphic to it. This group is isomorphic to the semidirect product of the abelian {3, 7}-group, which contains an element of order 21, by the symmetric group of degree 5.     

Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal; that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.     

On self-normality and abnormality in the alternating group Ap

An old problem proposed by Huppert, Doerk and Hawkes motivates us to investigate the relationship between an abnormal subgroup and self-normalizing in non-solvable groups. A subgroup H of a group G is called second maximal if H is maximal in all maximal subgroups of G containing H. Our result is that if H is a second maximal subgroup of the alternating group A_p of prime degree, then H is abnormal in A_p if and only if H is self-normalizing.     

Graph puzzles,homotopy, and the alternating group

The so-called 15-puzzle may be generalized to a puzzle based on an arbitrary graph. We consider labelings or colorings of the vertices and the operation of switching one distinguished label with a label on an adjacent vertex. Starting from a given labeling, iterations of this operation allow one to obtain all, or exactly half, of the labelings on a non-separable graph (with the polygons and one other graph as exceptions).     

Maximal irredundant families of minimal size in the alternating group

Archiv der Mathematik - Let G be a finite group. A family $${mathcal {M}}$$ of maximal subgroups of G is called “irredundant” if its intersection is not equal to the intersection of...     

A normal generating set for the Torelli group of a compact non-orientable surface

For a compact surface S, let \({\mathcal {I}}(S)\) denote the Torelli group of S. For a compact orientable surface \(\Sigma \), \({\mathcal {I}}(\Sigma )\) is generated by two types of mapping classes, called bounding simple closed curve maps (BSCC maps) and bounding pair maps (BP maps) (see Powell in Proc Am Math Soc 68:347–350, 1978; Putman in Geom Topol 11:829–865, 2007). For a non-orientable closed surface N, \({\mathcal {I}}(N)\) is generated by BSCC maps and BP maps (see Hirose and Kobayashi in Fund Math 238:29–51, 2017). In this paper, we give an explicit normal generating set for \({\mathcal {I}}(N_g^b)\), where \(N_g^b\) is a genus-g compact non-orientable surface with b boundary components for \(g\ge 4\) and \(b\ge 1\).