On permutation characters of wreath products

It is known that the character rings of symmetric groups S_n and the character rings of hyperoctahedral groups S₂?S_n are generated by (transitive) permutation characters. These results of Young are generalized to wreath products G?H (G a finite group, H a permutation group acting on a finite set). It is shown that the character ring of G?H is generated by permutation characters if this holds for G, H and certain subgroups of H. This result can be sharpened for wreath products G?S_n;if the character ring of G has a basis of transitive permutation characters, then the same holds for the character ring of G?S_n.     

Nil-automorphisms of groups with residual properties

Let G be any group and x an automorphism of G. The automorphism x is said to be nil if, for every g ∈ G, there exists n = n(g) such that [g, _n x] = 1. If n can be chosen independently of g, we say that x is n-unipotent. A nil (resp. unipotent) automorphism x could also be seen as a left Engel element (resp. left n-Engel element) in the group G〈x〉. When G is a finite dimensional vector space, groups of unipotent linear automorphisms turn out to be nilpotent, so that one might ask to what extent this result can be extended to a more general setting. In this paper we study finitely generated groups of nil or unipotent automorphisms of groups with residual properties (e.g. locally graded groups, residually finite groups, profinite groups), proving that such groups are nilpotent.     

A subgroup involvement of the Fibonacci length

For a non-Abelian 2-generated finite group G=〈a,b〉, the Fibonacci length of G with respect to A={a,b}, denoted by LEN _A (G), is defined to be the period of the sequence x ₁=a,x ₂=b,x ₃=x ₁ x ₂,…,x _n+1=x _n?1 x _n ,… of the elements of G. For a finite cyclic group C _n =〈a〉, LEN _A (C _n ) is defined in a similar way where A={1,a} and it is known that LEN _A (C _n )=k(n), the well-known Wall number of n. Over all of the interesting numerical results on the Fibonacci length of finite groups which have been obtained by many authors since 1990, an intrinsic property has been studied in this paper. Indeed, by studying the family of minimal non-Abelian p-groups it will be shown that for every group G of this family, there exists a suitable generating set A′ for the derived subgroup G′ such that LEN _A′(G′)|LEN _A (G) where, A is the original generating set of G.     

Nilpotency of group ring units symmetric with respect to an involution

Let F be an infinite field of characteristic different from 2. Let G be a torsion group having an involution ∗, and consider the units of the group ring FG that are symmetric with respect to the induced involution. We classify the groups G such that these symmetric units satisfy a nilpotency identity (x₁,…,x_n)=1.     

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* _h (G) = G, where F* ₁ (G) = F* (G) is the generalized Fitting subgroup, and F* _i+1(G) is the inverse image of F* (G/F*_i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.     

Unitary graphs and classification of a family of symmetric graphs with complete quotients

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.     

A construction of orthogonal arrays and applications to embedding theorems

An n^t by k orthogonal array is a collection of k-tuples of elements from an n-set, such that if a matrix is formed with the k-tuples as rows then each ordered t-tuple of elements appears exactly once as a row of each t columned and n^t rowed submatrix. If such an array has its set of k-tuples invariant under the elements of a subgroup G of S_t then the array is referred to as a G-array. A method is described for constructing a G-array of order nr from an array of order n and G-arrays of order r.The above described construction is used to produce finite embedding theorems for partial 3-quasigroups of various types. For a class of 3-quasigroups, such a theorem shows that a finite partial member of the class can be embedded in a finite complete member of the class. Theorems included produce finite embedding theorems for 3-quasigroups satisfying the identities 〈x,y,〈y,x,z〉〉=z and 〈〈z,x,y〉,y,x〉=z, for cyclic 3-quasigroup s, and conditional embedding theorems are presented for semi-symmetric 3-quasigroups.     

Doubly transitive automorphism groups of block designs

If G is a doubly transitive group of automorphisms of a block design with λ = 1, then for any block Δ of the design and any point α in Δ, the set Δ?{α} is a block of imprimitivity for G_α. What are sufficient conditions for a doubly transitive but not doubly primitive permutation group G to be a group of automorphisms of a non-trivial block design with λ = 1 ? Can the design or the group G be identified if there is a nonidentity automorphism in G fixing every point of some block of the design? Both of these questions are investigated and some answers are given.     

Automorphism groups of linear spaces and their parabolic subgroups

Suppose that an almost simple group G acts line transitively on a finite linear space S. Let Gx be a point stabilizer in G and suppose that G has socle T, a simple group of Lie type. In this paper we show that if T∩Gx is a parabolic subgroup of T, then G is flag transitive on S.     

Finite generation of iterated wreath products

Let (G _n , X _n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product ${\ldots\wr G_2\wr G_1}Let (G _n , X _n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product ?\wr G₂\wr G₁{\ldots\wr G_2\wr G_1} is topologically finitely generated if and only if the profinite abelian group ?_{n 3 1} G_n/G￠_n{\prod_{n\geq 1} G_n/G'_n} is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power G\wr ?\wr G\wr G{G\wr \ldots\wr G\wr G} (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.     

Achievement and avoidance games for generating abelian groups

For any finite groupG, the DO GENERATE game is played by two players Alpha and Beta as follows. Alpha moves first and choosesx ₁ ∈G. Thek-th play consists of a choice ofx _k ∈G ?S _k ?1 whereS _n ={itx ₁,...,x _n }. LetG _n = 〈S _n 〉. The game ends whenG _n =G. The player who movesx _n wins. In the corresponding avoidance game, DON'T GENERATE, the last player to move loses. Of course neither game can end in a draw. For an arbitrary group, it is an unsolved problem to determine whether Alpha or Beta wins either game. However these two questions are answered here for abelian groups.     

Extension of Multiply Transitive Permutation Sets

Let G be a k-transitive permutation set on E and let E* = E∪{∞},∞ ? E; if G* is a (k: + 1)-transitive permutation set on E*, G* is said to be an extension of G whenever G _∝ ^* =G. In this work we deal with the problem of extending (sharply) k- transitive permutation sets into (sharply) (k + 1)-transitive permutation sets. In particular we give sufficient conditions for the extension of such sets; these conditions can be reduced to a unique one (which is a necessary condition too) whenever the considered set is a group. Furthermore we establish necessary and sufficient conditions for a sharply k- transitive permutation set (k ≥ 3) to be a group. Math. Subj. Class.: 20B20 Multiply finite transitive permutation groups 20B22 Multiply infinite transitive permutation groups     

On difference sets in high exponent 2-groups

We investigate the existence of difference sets in particular 2-groups. Being aware of the famous necessary conditions derived from Turyn’s and Ma’s theorems, we develop a new method to cover necessary conditions for the existence of (2^2d+2,2^2d+1?2 ^d ,2^2d ?2 ^d ) difference sets, for some large classes of 2-groups. If a 2-group G possesses a normal cyclic subgroup 〈x〉 of order greater than 2 ^d+3+p , where the outer elements act on the cyclic subgroup similarly as in the dihedral, semidihedral, quaternion or modular groups and 2 ^p describes the size of G′∩〈x〉 or C _G (x)′∩〈x〉, then there is no difference set in such a group. Technically, we use a simple fact on how sums of 2 ⁿ -roots of unity can be annulated and use it to characterize properties of norm invariance (prescribed norm). This approach gives necessary conditions when a linear combination of 2 ⁿ -roots of unity remains unchanged under homomorphism actions in the sense of the norm.     

Maximal subgroups and PST-groups

A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.     

The group PSL(2,q) is not the multiplication group of a loop

ABSTRACT

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ?, ψ ∈ End(G) such that x? = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.     

The k-factor conjecture is true

A sequence 〈d_i〉, 1≤i≤n, is called graphical if there exists a graph whose i^th vertex has degree d_i for all i. It is shown that the sequences 〈d_i〉 and 〈d_i-k〉 are graphical only if there exists a graph G whose degree sequence is 〈d_i〉 and which has a regular subgraph with k lines at each vertex. Similar theorems have been obtained for digraphs. These theorems resolve comjectures given by A.R. Rao and S.B. Rao, and by B. Grünbaum.     

On the exponentiality of affine symmetric spaces

An affine symmetric space G/H is said to be exponential if every two points of this space can be joined by a geodesic and weakly exponential if the union of all geodesics issuing from one point is everywhere dense in G/H. For the group space (G × G)/G _diag of a Lie group G, these properties are equivalent to the exponentiality and weak exponentiality of G, respectively. We generalize known theorems on the image of the exponential mapping in Lie groups to the case of affine symmetric spaces. We prove the weak exponentiality of the symmetric spaces of solvable Lie groups, and in the semisimple case we obtain criteria for exponentiality and weak exponentiality.     

FINITE PERMUTATION REPRESENTATION OF A SUBGROUP OF PICARD GROUP＊

We investigate action of a subgroup G1 of the Picard group on finite sets using coset diagrams.We show that its actions on the sets of 3,4,5,6,8,and 12 elements yield building blocks of Coset diagrams ...     

Multiple existence of non-radial solutions with group invariance for sublinear elliptic equations

We deal with sublinear elliptic equations in a ball and prove the existence of infinitely many solutions which are not radially symmetric but G invariant. Here G is any closed subgroup of the orthogonal group and is not transitive on the unit sphere.