algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation)...

subgroup of the group other than the entire group Generating set of a ring: A subset S of a ring A generates A if the only subring of A containing S is A Generating...

} A subset S of a group G is called a generating set if every element of G can be represented by a word in S. When S is not a generating set for G...

of group theory, a strong generating set of a permutation group is a generating set that clearly exhibits the permutation structure as described by a...

group cannot be free, since the elements of a free generating set of a free group have infinite order. In algebraic topology, the fundamental group of...

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...

. In this case, we say that the set { x 1 , … , x s } {\displaystyle \{x_{1},\dots ,x_{s}\}} is a generating set of G {\displaystyle G} or that x 1 ...

constructing expander graphs. Let G {\displaystyle G} be a group and S {\displaystyle S} be a generating set of G {\displaystyle G} . The Cayley graph Γ = Γ ( G...

Generator matrix, a matrix used in coding theory Generator (category theory) of a category, in category mathematics Generating set of a group, group generators...

multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert...

of a free module Rank of a greedoid, the maximal size of a feasible set Rank of a group, the smallest cardinality of a generating set for the group Rank...

a matroid Generating set of an ideal: Gröbner basis Hilbert's basis theorem Generating set of a group Base (topology) Change of basis Greedoid Normal...

expressed as a word whose letters come from a generating set for the group. The word metric on G is very closely related to the Cayley graph of G: the word...

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 the generating set from its presentation as a Coxeter group. Such a generating set for a dihedral group of order 10 consists of any pair of elements...

mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every...

Free abelian group Free group Free product Generating set of a group Group cohomology Group extension Presentation of a group Product of group subsets Schur...

is, the collection of all open sets in X . {\displaystyle X.} There are multiple (non-equivalent) definitions of compactly generated space or k-space in...

In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one...

with involution on A {\displaystyle A} to the group G {\displaystyle G} . If B {\displaystyle B} is another finite generating set for G {\displaystyle...

mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite...

(2p^{k})=p^{k}-p^{k-1}.} By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers g 0 ,...

their symmetries. The set of symmetries of a frieze pattern is called a frieze group. Frieze groups are two-dimensional line groups, having repetition in...

a presentation is one method of specifying a group. A presentation of a group G comprises a set S of generators—so that every element of the group can...

the commutative ring of p-adic numbers), that is generated by a single element. That is, it is a set of invertible elements with a single associative binary...

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation)...

forming a sort of dendritic network. At the roots of the theorem-space are found the axioms, which "generate" the theorem-space much like a generating set generates...

symmetric group with the structure of a Coxeter group (and so also a reflection group). Other possible generating sets include the set of transpositions...

set S generates G, so a boundedly generated group is finitely generated. An equivalent definition can be given in terms of cyclic subgroups. A group G...

groups, and briefly mentioned (on page 274) the group M24, giving its order. In Mathieu (1873) he gave further details, including explicit generating...