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The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract 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 finitely many elements of the subset and their inverses.
Generator (mathematics) The 5th roots of unity in the complex plane under multiplication form a group of order 5. Each non-identity element by itself is a generator for the whole group. In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that ...
Presentation of a group. In mathematics, 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 be written as a product of powers of some of these generators—and a set R of relations among those generators. We then say G has presentation.
An abstract group defined by this multiplication is often denoted C n, and we say that G is isomorphic to the standard cyclic group C n. Such a group is also isomorphic to Z / n Z , the group of integers modulo n with the addition operation, which is the standard cyclic group in additive notation.
Abstract syntax tree. An abstract syntax tree (AST) is a data structure used in computer science to represent the structure of a program or code snippet. It is a tree representation of the abstract syntactic structure of text (often source code) written in a formal language. Each node of the tree denotes a construct occurring in the text.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group S n {\displaystyle \mathrm {S} _{n}} defined over a finite set of n {\displaystyle n} symbols consists of ...
Commutator subgroup. In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. [1][2] The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this ...
Free module. In mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, [1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.