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)...
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Generator (mathematics) (redirect from Generating set)
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...
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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...
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} 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...
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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...
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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...
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. 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 ...
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Cayley graph (category Group theory)
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...
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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...
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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...
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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...
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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...
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Word metric (category Properties of groups)
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...
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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...
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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...
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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...
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with involution on A {\displaystyle A} to the group G {\displaystyle G} . If B {\displaystyle B} is another finite generating set for G {\displaystyle...
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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...
19 KB (2,837 words) - 22:51, 25 November 2023
(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 ,...
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countably generated can have several meanings: An algebraic structure (group, module, algebra) having countably many generators, see generating set Countably...
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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)...
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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...
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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...
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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...
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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...
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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,...
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symmetric group with the structure of a Coxeter group (and so also a reflection group). Other possible generating sets include the set of transpositions...
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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...
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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...
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