On Bergman’s Property for the Automorphism Groups of Relatively Free Groups
نویسنده
چکیده
This easily implies that the confinality of Sym(Ω) is greater than |Ω|, the result first proven by Macpherson and Neumann in [15] (the confinality of a given groupG being the least cardinality of a chain of proper subgroup whose union is G.) Another consequence of the result of Bergman’s is really surprising! One of the examples of systems (Yi) is given by the system of powers (X : m ∈ N) of a generating set X of Sym(Ω) which is closed under taking inverses. By a power X of X we mean the set of all products of the form x1 . . . xm, where xi ∈ X. Then it follows from (1.1) that Sym(Ω) equals X for some natural number k. This naturally leads to the following definition.
منابع مشابه
Infinite-dimensional General Linear Groups Are Groups of Universally Finite Width
Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the universality of finite width: given any generating set X of the symmetric group of an infinite set Ω, there is a uniform bound k such that any permutation σ ∈ Sym(Ω) is a product of at most k elements of X∪X. Bergman also formulated a sort of general conjecture statin...
متن کاملNILPOTENCY AND SOLUBILITY OF GROUPS RELATIVE TO AN AUTOMORPHISM
In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...
متن کاملElementary Properties of Cycle-free Partial Orders and their Automorphism Groups
1. Abstract A classiication was given in 1, 12, 13] of all the countable k-CS-transitive cycle-free partial orders for k 3. Here the elementary theories of these structures and their automorphism groups are examined, and it is shown that in many cases we can distinguish the structures or their groups by means of their rst or second order properties. The small index property is established for w...
متن کاملCofinitely Hopfian Groups, Open Mappings and Knot Complements
A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivi...
متن کاملOn the nilpotency class of the automorphism group of some finite p-groups
Let $G$ be a $p$-group of order $p^n$ and $Phi$=$Phi(G)$ be the Frattini subgroup of $G$. It is shown that the nilpotency class of $Autf(G)$, the group of all automorphisms of $G$ centralizing $G/ Fr(G)$, takes the maximum value $n-2$ if and only if $G$ is of maximal class. We also determine the nilpotency class of $Autf(G)$ when $G$ is a finite abelian $p$-group.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008