On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

Representing the automorphism group of an almost crystallographic group

Let be an almost crystallographic (AC-) group, corresponding to the simply connected, connected, nilpotent Lie group and with holonomy group . If , there is a faithful representation . In case is crystallographic, this condition is known to be equivalent to or . We will show (Example 2.2) that, for AC-groups , this is no longer valid and should be adapted. A generalised equivalent algebraic (and easier to verify) condition is presented (Theorem 2.3). Corresponding to an AC-group and by factoring out subsequent centers we construct a series of AC-groups, which becomes constant after a finite number of terms. Under suitable conditions, this opens a way to represent faithfully in (Theorem 4.1). We show how this can be used to calculate . This is of importance, especially, when is almost Bieberbach and, hence, is known to have an interesting geometric meaning.

     

Automorphism group and diameter of a graph

Given a connected graph Γ of order n and diameter d, we establish a tight upper bound for the order of the automorphism group of Γ as a function of n and d, and determine the graphs for which the bound is attained. © 2011 Wiley Periodicals, Inc. J Graph Theory.     

On the isomorphism problem in the class of extensions of a free group by an infinite cyclic group

In the automorphism group of a rank 3 free group we find two nonconjugate automorphisms whose mapping tori are isomorphic groups.     

On local automorphisms of group algebras of compact groups

We show that with few exceptions every local isometric automorphism of the group algebra of a compact metric group is an isometric automorphism.

     

Automorphism group and representation of a twisted multi-loop algebra

Let G be the complexification of the real Lie algebra so（3） and A = C[t1^±1, t2^±1] be the Lau-ent polynomial algebra with commuting variables. Let L：（t1, t2, 1） = G c .A be the twisted multi-loop Lie algebra. Recently we have studied the universal central extension, derivations and its vertex operator representations. In the present paper we study the automorphism group and bosonic representations ofL（t1, t2, 1）.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .

Theorem. If is an automorphism of , then there is an such that for all .

     

On extendability of group actions on compact Riemann surfaces

The question of whether a given group which acts faithfully on a compact Riemann surface of genus is the full group of automorphisms of (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group in terms of a concrete partial presentation for associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where is a non-cyclic abelian group.

     

The probability of generating a finite simple group

We show that two random elements of a finite simple groupG generateG with probability 1 as |G| . This settles a conjecture of Dixon.     

换位子群为p阶群的有限p-群的自同构群

设G是换位子群为p阶群的有限p-群,确定了AutG的结构,证明了(i)AutG/AutGG≌Zp-1,其中AutGG={α∈AutG|α平凡地作用在G上}.(ii)AutGG/Op(AutG)≌iGL(ni,p)×jSp(2mj,p),其中Op(AutG)是AutG的最大正规p-子群,ni和mj由G惟一确定.     

The probability of generating a finite simple group

We study the probability of generating a finite simple group, together with its generalisation P _G,socG (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/socG. We prove that P _G,socG (2) ? 53/90, with equality if and only if G is A₆ or S₆, and establish a similar result for P _G,socG (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.     

The probability of generating a finite linear group

We study the asymptotic behavior of the probability of generating a finite completely reducible linear group G of degree n with [ n] elements. In particular we prove that if 3/2 and n is large enough then [ n] randomly chosen elements that generate G modulo O²(G) almost certainly generate G itself.Received: 13 February 2003     

The probability of generating a finite classical group

In this paper generalized Fischer spaces are defined and some results on their classification are obtained.     

Symmetric designs with parameters (69, 17, 4) and F39 as a group of automorphisms

According to Mathon and Rosa [The CRC handbook of combinatorial designs, CRC Press, 1996] there is only one known symmetric design with parameters (69, 17, 4). This known design is given in Beth, Jungnickel, and Lenz [Design theory, B. I. Mannheim, 1985]; the Frobenius group F₃₉ of order 39 acts on this design, where Z₁₃ has exactly 4 fixed points and Z₃ has exactly 9 fixed points. The purpose of this article is to investigate the converse of this fact with the hope of obtaining in this way at least one more design with these parameters. In fact we obtain exactly one new such design. In this article we have classified all such designs invariant under F₃₉. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 231–233, 1998     

Smallest graphs with given generalized quaternion automorphism group

For , a smallest graph whose automorphism group is isomorphic to the generalized quaternion group is constructed. If , then such a graph has vertices and edges. In the special case when , a smallest graph has 16 vertices but 44 edges.