Finite quotients of the automorphism group of a free group

The automorphism group AutF_n of a free group F_n of rank n acts on the product of n copies of a group G by substituting n elements of G into the words defining an automorphism of the free group. This gives rise to an antihomomorphism from AutF_nto a permutation group. We determine this antihomomorphic image of AutF_n when G is the semidirect product Z_p x Z_q     

Generic polynomials for transitive subgroups of the group S 4 over fields of characteristic 2

Generic polynomials for the symmetric group S₄, the Klein group V₄, the cyclic group of order 4, the dihedral group D₄, and the alternating group A₄ over fields of characteristic 2 are described explicitly. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 247–258.     

Group of automorphisms of the ring ℤA4

The structure of the group of automorphisms of the integer group ring of the group A₄ is studied in terms of a semidirect product. We show that the Zassenhaus conjecture on the structure of automorphisms of integer group rings of finite groups for the group Aut ?A₄ holds.     

Double centralizing theorem for wreath products of the alternating group

The double centralizing theorem between the action of the symmetric group S_n and the action of the general linear group on the tensor space T_n(W) was obtained by Schur. Here we obtain a double centralizing theorem when S_n is replaced by the wreath product of a finite group G and the alternating group A_n.     

Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

Complex-valued preplaces and the nonring cmc subsets of a ring

We show that the torsion free Krull-Schmidt-Azumaya Theorem holds for the integral group ring Z D ₈ Were D ₈ is the group of order 8, but torsion free cancellation (and hence also the torsion free Krull-Schmidt-Azumaya theorem) fails for the integral group ring Z D ₃₂ We summarize the cases for which the integral group ring Z G is known to satisfy the torsion free Krull-Schmidt-Azumaya Theorem, Z D ₁₆ being the only open case.     

On Finite Simple Groups with the Set of Element Orders as in a Frobenius Group or a Double Frobenius Group

It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to L₃(3) or U₃(3) (to U₃(3) or S₄(3), respectively).     

Graphs with symmetric automorphism group

We investigate the properties of graphs whose automorphism group is the symmetric group. In particular, we characterize graphs on less than 2n points with group S_n, and construct all graphs on n + 3 points with group S_n. Graphs with 2n or more points and group S_n are discussed briefly.     

On the Constructive Inverse Problem in Differential Galois Theory#

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G ⁰ = G ₁ · ··· · G _r , where each G _i is a simple group of type A_?, C_?, D_?, E₆, or E₇, we construct a differential equation over C(x) having Galois group G.     

Minimal translation surfaces in the Heisenberg group Nil3

A translation surface in the Heisenberg group Nil₃ is a surface constructed by multiplying (using the group operation) two curves. We completely classify minimal translation surfaces in the Heisenberg group Nil₃.     

Integral Group Ring of the Mathieu Simple Group M 23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M ₂₃ using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M ₁₁ and M ₁₂. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.     

A Semiring Whose Grothendieck Group is not Isomorphic to the Grothendieck Group of the Ring completion of the Semiring

We define the Grothendieck group of a semiring and present an example of the Grothendieck group K₀R of a semiring R such that K₀R is not isomorphic to the Grothendieck group of the ring completion of R.     

The unit group of FS 4

A complete characterization is given for the unit group U(FS ₄) of the group algebra FS ₄ of the symmetric group S ₄ of degree 4 over a finite field F.      

On automorphisms of finite simple Chevalley groups

The paper deals with the splitting properties of the automorphism groups of finite Chevalley groups. Using the action of symmetries of a Dynkin diagram on the corresponding Weyl group, a sufficient condition is developed for the existence of a complement for the inner automorphism group in the automor ph-ism group of a finite Chevalley group. The condition is verified for Chevalley groups of the classical types (viz., the group of type A_l, B_l, C_l,and D_l) as well as the exceptional groups of type E₆ and E₇, under suitable restrictions on the base fields.     

Explicit determination of generalized symmetric and alternating Galois groups

This paper shows a probabilistic algorithm to decide whether the Galois group of a given irreducible polynomial with rational coefficients is the generalized symmetric group C_p?S_m or the generalized alternating group C_p?A_m. In the affirmative case, we give generators of the group with their action on the set of roots of the polynomial.     

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

   Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

RELATIVE INJECTIVITY OF MODULES AND EXCELLENT EXTENSIONS

Abstract

Yesl Any equation of conservation of the form ?_x{P(?_xφ, ?_tφ) = ?_t{Q(?_xφ, ?_tφ) is shown to admit an infinite-dimensional, Abellan group of symmetries that is not a prolongation symmetry group. Explicit equations are given for the determination of the generators of the Lle algebra of this Abellan symmetry group, and for the generators of Its underlying Poisson algebra.     

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ?₃ × ?_{3 ^k} or ?₃ × ?₃ × ? _p where k ≥ 1 and p is a prime. In addition, we prove that ?₂ × ?₂ × ? _p is a Schur group for every prime p.