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Concept
Finitely generated group
Summary
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements.
By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
Every quotient of a finitely generated group G is finitely generated; the quotient group is generated by the images of the generators of G under the canonical projection.
A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z.
A locally cyclic group is a group in which every finitely generated subgroup is cyclic.
The free group on a finite set is finitely generated by the elements of that set (§Examples).
A fortiori, every finitely presented group (§Examples) is finitely generated.
Finitely generated abelian group
Every abelian group can be seen as a module over the ring of integers Z, and in a finitely generated abelian group with generators x1, ..., xn, every group element x can be written as a linear combination of these generators,
x = α1⋅x1 + α2⋅x2 + ... + αn⋅xn
with integers α1, ..., αn.
Subgroups of a finitely generated abelian group are themselves finitely generated.
The fundamental theorem of finitely generated abelian groups states that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of which are unique up to isomorphism.
A subgroup of a finitely generated group need not be finitely generated. The commutator subgroup of the free group on two generators is an example of a subgroup of a finitely generated group that is not finitely generated.
On the other hand, all subgroups of a finitely generated abelian group are finitely generated.
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Related publications (1)
Moyennabilité et courbure
As Avez showed (in 1970), the fundamental group of a compact Riemannian manifold of nonpositive sectional curvature has exponential growth if and only if it is not flat. After several generalizations from Gromov, Zimmer, Anderson, Burger and Shroeder, the following theorem was proved by Adams and Ballmann (in 1998). Theorem Let X be a proper CAT(0) space. If Γ is an amenable group of isometries of X, then at least one of the following two assertions holds: Γ fixes a point in ∂X (boundary of X). X contains a Γ-invariant flat (isometric copy of Rn, n ≥ 0). Following an idea of my PhD advisor Nicolas Monod, I tried to generalize this theorem in the context of goupoids, in this case Borel G-spaces and countable Borel equivalence relations. This lead me to study the notion of Borel fields of metric spaces, which turns out to be a suitable context to define an action of a countable Borel equivalence relation. A field of metric spaces over a set Ω is a family {(Xω,dω)} ω∈Ω of nonempty metric spaces denoted by (Ω,X•). We introduced as S( Ω,X•) the set of maps Such maps are called sections. If Ω is a Borel space, we can define a Borel structure on a field of metric spaces to be a subset Lℒ( Ω,X•) of S( Ω,X•) satisfying these three conditions For all f, g ∈ ℒ(Ω,X•), the function Ω → R, ω → dω(f(ω), g(ω)) is Borel. If h ∈ S(Ω,X•) is such that the function Ω → R, ω → dω(f(ω), h(ω)) is Borel for all f ∈ ℒ(Ω,X•), then h ∈ ℒ(Ω,X•). There exists a countable family of sections {fn}n≥1 ⊆ ℒ(Ω,X•) such that {fn (ω)}n≥1 = Xω for all ω ∈ Ω. This definition is consistent with more classical definitions of Borel fields of Banach spaces or of Borel fields of Hilbert spaces. The notion of a Borel field of metric spaces has been used in convex analysis and in economy. As said before, we can define an action of a countable Borel equivalence relation ℛ ⊆ Ω2 on a Borel field of metric spaces (Ω,X•) in a natural way. It's determined by a family of bijectives maps {α(ω, ω') : Xω → Xω'}(ω,ω')∈ℛ such that For all (ω,ω'), (ω',ω") ∈ ℛ the following equality is satisfied α(ω', ω") ◦ α(ω, ω') = α(ω, ω"). For all f, g ∈ ℒ(Ω,X), the function ℛ → R, (ω, ω') → dω(f(ω), α(ω', ω)g(ω')) is Borel. Zimmer (1977) introduced the notion of amenability for ergodic G-spaces and equivalence relations, of which we obtained the first generalization (in collaboration with Philippe Henry). Theorem Let R be a countable, Borel, preserving the class of the measure, ergodic and amenable equivalence relation on the probability space Ω acting on a Borel field ( Ω,X•) of proper CAT(0) spaces with finite topological dimension. Then at least one of the following assertions is true: There exists an ℛ-invariant Borel section ξ ∈ L(Ω,∂X•). There exists an ℛ-invariant Borel subfield (Ω, F•) of (Ω,X•) consisting of flat subsets. And the second generalization for amenable ergodic G-spaces. Theorem Let G be a locally compact second countable group, Ω a preserving class of the measure, ergodic amenable G-space, X a proper CAT(0) space with finite topological dimension and α : G × Ω → Iso(X) a Borel cocycle. Then at least one of the following assertions is true: There exists an α-invariant Borel function ξ : Ω → ∂X. There exists an α-invariant borelian subfield (Ω, F•) of the trivial field (Ω, X) consisting of flat subsets. If we consider (Ω,μ) to be a strong boundary of the group G, the cocycle α to come from an action of G on X, and X to have flats of at most dimension 2, then we can conclude the following. Theorem Let G be a locally compact second countable group, (Ω,μ) a strong boundary of G, X a proper CAT(0) space with finite topological dimension and whose flats are of dimension at most 2. Let suppose that G acts by isometry on X. Then at least one of the following assertions is true: There exists a G-equivariant Borel function ξ: Ω → ∂X. There exists a G-invariant flat F in X. The proof of the three theorems are strongly based on properties of Borel field of metric spaces that we prove in this thesis.
EPFL2010Related concepts (35)
Finitely generated group
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of S and of inverses of such elements. By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
Torsion (algebra)
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element. This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements.
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
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