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)...

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...

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...

} 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...

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...

. 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 ...

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...

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

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...

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...

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...

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...

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...

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

elements. A very important generating set of a dihedral group is the generating set from its presentation as a Coxeter group. Such a generating set for a dihedral...

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...

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...

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...

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

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...

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...

subgroups generated by a subset of the special generating set. Separately, the symmetric group belongs to a larger family of reflection groups called complex...

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

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...

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...

(relatively prime) to n from the set { 0 , 1 , … , n − 1 } {\displaystyle \{0,1,\dots ,n-1\}} of n non-negative integers form a group under multiplication modulo...

Computer-generated imagery (CGI) is a specific-technology or application of computer graphics for creating or improving images in art, printed media, simulators...

given a set of generators for the group. Cycle graphs are particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers...

of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940...