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...

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...

} 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...

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...

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 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...

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...

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...

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...

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...

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...

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...

addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence of term coefficients...

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...

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 ,...

countably generated can have several meanings: An algebraic structure (group, module, algebra) having countably many generators, see generating set Countably...

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)...

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...

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...

mathematical subject of group theory, the rank of a group G, denoted rank(G), can refer to the smallest cardinality of a generating set for G, that is rank...

that is, a group with a finite number of elements. finitely generated group A group G is finitely generated if there is a finite generating set, that is...

group of order s, the Cayley graph for a generating set with one generator is an s-vertex cycle graph. The diameter of this graph, and of the group,...

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

hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological...

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...