Divisibility in certain automorphism groups

We study solvability of equations of the form x ⁿ = g in the groups of order automorphisms of archimedean-complete totally ordered groups of rank 2. We determine exactly which automorphisms of the unique abelian such group have square roots, and we describe all automorphisms of the general ones.     

Residual finiteness of outer automorphism groups of certain 1-relator groups

We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form a,b;(a-αbβaαbγ)t , t 1, having residually finite outer automorphism groups.     

Kazhdan groups with infinite outer automorphism group

For each countable group we produce a short exact sequence where has a graphical presentation and is f.g. and satisfies property .

As a consequence we produce a group with property  such that is infinite.

Using the tools developed we are also able to produce examples of non-Hopfian and non-coHopfian groups with property .

One of our main tools is the use of random groups to achieve certain properties.

     

Symmetry groups of some perfect 1-factorizations of complete graphs

The following problem has arisen in the study of graphs, lattices and finite topologies. Is there a 1-factorization of K_2m the complete graph on 2n points, such that the union of every pair of distinct 1-factors is a hamiltonian circuit? In this paper it is noted that on $\text{K}{}_{\mathrm{2m}}\text{1?n?5}$, there is, up to relabelling, only one 1-factorization of the required type. On K₁₂ and whenever there are odd primes p,q>3 such that p + 1 = 2q, there are at least two different such 1-factorizations. These results are obtained by computing symmetry groups. The symmetry groups obtained are Frobenius groups of maximal order (i.e., sharply 2-transitive groups) and direct products of these groups with the group of order 2.     

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 We will show (Example 2.2) that, for AC-groups , this is no longer valid and should be adapted.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.

     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

Rainbow spanning tree decompositions in complete graphs colored by cyclic 1-factorizations

Brualdi and Hollingsworth conjectured in Brualdi and Hollingsworth (1996) that in any complete graph $K_{2n}$, $n\ge 3$, which is properly colored with $2n?1$ colors, the edge set can be partitioned into $n$ edge disjoint rainbow spanning trees (where a graph is said to be rainbow if its edges have distinct colors). Constantine (2002) strengthened this conjecture asking the rainbow spanning trees to be pairwise isomorphic. He also showed an example satisfying his conjecture for every $2n\in \{2^s:s\ge 3\}\cup \{5?2^s,s\ge 1\}$ . Caughmann, Krussel and Mahoney (2017) recently showed a first infinite family of edge colorings for which the conjecture of Brualdi and Hollingsworth can be verified. In the present paper, we extend this result to all edge-colorings arising from cyclic 1-factorizations of $K_{2n}$ constructed by Hartman and Rosa (1985). Finally, we remark that our constructions permit to extend Constatine’s result also to all $2n\in \{2^{s}d:s\ge 1,d>3\text{odd}\}$.     

Endomorphisms of automorphism groups of free groups

It is proved that any non-trivial endomorphism of an automorphism group AutF_n of a free group F_n, for n 3, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of AutF₂ is an automorphism, or else a homomorphism onto one of the groups S₃, D₈, Z₂ × Z₂, Z₂, or (Z₂ × Z₂). A non-trivial homomorphism of AutF_n into AutF_m, for n 3, m 2, and n > m, is a homomorphism onto Z₂ with kernel SAutF_n. As a consequence, we obtain that AutF_n is co-Hopfian.Supported by RFBR grant No. 02-01-00293 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 211–237, March–April, 2005.     

Geometric automorphism groups of graphs

Constructing symmetric drawings of graphs is NP-hard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define an n-geometric automorphism group as a subgroup of the automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2- and 3-geometric automorphism groups of a given graph. We implement the algorithm using Magma [〈http://magma.maths.usyd.edu.au〉] and the experimental results show that our approach is very efficient in practice. We also present a drawing algorithm to display 2- and 3-geometric automorphism groups.     

On outer automorphism groups of coxeter groups

Summary It is shown that the outer automorphism group of a Coxeter groupW of finite rank is finite if the Coxeter graph contains no infinite bonds. A key step in the proof is to show that if the group is irreducible andΠ ₁ andΠ ₂ any two bases of the root system ofW, thenΠ ₂ = ±ωΠ ₁ for some ω εW. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Brink and Howlett. This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

On the upper-semicontinuity of automorphism groups of model domains in almost complex manifolds

We will prove that the automorphism groups of the strongly pseudoconvex model domains in almost complex manifolds are isomorphically embedded into the automorphism group of the unit ball.     

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.     

Structure of weak automorphism groups of mono-unary algebras

A weak automorphism of an algebra on A with the set T of term operations is a permutation of of A such that . In this note we describe the groups of weak automorphisms of mono-unary algebras. Some examples of such algebras without weak automorphisms are given. Received January 31, 1996; accepted in final form March 16, 1999.     

Endomorphism rings and automorphism groups of almost completely decomposable groups

To every non-cuspidal K-rational point on the modular curve X_l(n) a non-commutative Noetherian domain of global dimension 3 can be associated : the Sklyanin algebra. In this paper we give the defining equations of the Sklyanin algebras and their centers when X_l(n) is rational, i.e. n <= 10 or n = 12.     

Finite groups with its power automorphism groups having small indices

In this paper, a finite group G with IAut（G） ： P（G）I ~- p or pq is determined, where P（G） is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies ｜Aut（G） ： P（G）｜= pq if and only if Aut（G）/P（G） ≌S3. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results.     

Characterization of automorphism groups of sporadic simple groups

It is proved that all automorphism groups of the sporadic simple groups are characterized by their element orders and the group orders.     

Direct products of automorphism groups of graphs

In this article we study the product action of the direct product of automorphism groups of graphs. We generalize the results of Watkins [J. Combin Theory 11 (1971), 95–104], Nowitz and Watkins [Monatsh. Math. 76 (1972), 168–171] and W. Imrich [Israel J. Math. 11 (1972), 258–264], and we show that except for an infinite family of groups S_n × S_n, n≥2 and three other groups D₄ × S₂, D₄ × D₄ and S₄ × S₂ × S₂, the direct product of automorphism groups of two graphs is itself the automorphism group of a graph. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 26–36, 2009