Thin group
Encyclopedia
In mathematics
, in the realm of group theory
, a group
is said to be thin if there is a finite upper bound on the girth of the Cayley graph
induced by any finite generating set
. The group is called fat if it is not thin.
Given any generating set of the group, we can consider a graph
whose vertices are elements of the group with two vertices adjacent if their ratio is in the generating set. The graph is connected
and vertex transitive. paths
in the graph correspond to words in the generators.
If the graph has a cycle
of a given length, it has a cycle of the same length containing the identity element. Thus, the girth of the graph corresponds to the minimum length of a nontrivial word that reduces to the identity. A nontrivial word is a word that, if viewed as a word in the free group, does not reduce to the identity.
If the graph has no cycles, its girth is set to be infinity.
The girth depends on the choice of generating set. A thin group is a group where the girth has an upper bound for all finite generating sets.
Some facts about thin and fat groups and about girths:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the realm of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is said to be thin if there is a finite upper bound on the girth of the Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
induced by any finite generating set
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
. The group is called fat if it is not thin.
Given any generating set of the group, we can consider a graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
whose vertices are elements of the group with two vertices adjacent if their ratio is in the generating set. The graph is connected
Connectivity (graph theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related to the theory of network flow problems...
and vertex transitive. paths
Path (graph theory)
In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...
in the graph correspond to words in the generators.
If the graph has a cycle
Cycle (graph theory)
In graph theory, the term cycle may refer to a closed path. If repeated vertices are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle, circuit, circle,...
of a given length, it has a cycle of the same length containing the identity element. Thus, the girth of the graph corresponds to the minimum length of a nontrivial word that reduces to the identity. A nontrivial word is a word that, if viewed as a word in the free group, does not reduce to the identity.
If the graph has no cycles, its girth is set to be infinity.
The girth depends on the choice of generating set. A thin group is a group where the girth has an upper bound for all finite generating sets.
Some facts about thin and fat groups and about girths:
- Every finite groupFinite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
is thin. - Every free groupFree groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
is fat. - The girth of a cyclic groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
equals its order. - The girth of a noncyclic abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
is at most 4, because any two elements commute and the commutation relation gives a nontrivial word. - The girth of the dihedral groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
is 2. - Every nilpotent groupNilpotent groupIn mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
, and more generally, every solvable groupSolvable groupIn mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
, is thin.