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Residue-class-wise affine group

Index Residue-class-wise affine group

In mathematics, specifically in group theory, residue-class-wise affine groups are certain permutation groups acting on \mathbb (the integers), whose elements are bijective residue-class-wise affine mappings. [1]

35 relations: Affine transformation, Bijection, Collatz conjecture, Computational group theory, Countable set, Cyclic group, Decision problem, Direct product of groups, Disjoint sets, Finite group, Fixed point (mathematics), Free group, Free product, Generating set of a group, Group (mathematics), Group action, Group theory, Injective function, Integer, Map (mathematics), Mathematics, Mathieu group, Permutation, Permutation group, Presentation of a group, Prime number, Restriction (mathematics), Ring (mathematics), Set (mathematics), Simple group, Subgroup, Surjective function, Uncountable set, Undecidable problem, Wreath product.

Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

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Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

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Collatz conjecture

The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term.

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Computational group theory

In mathematics, computational group theory is the study of groups by means of computers.

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Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

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Cyclic group

In algebra, a cyclic group or monogenous group is a group that is generated by a single element.

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Decision problem

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values.

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Direct product of groups

In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.

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Disjoint sets

In mathematics, two sets are said to be disjoint sets if they have no element in common.

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Finite group

In abstract algebra, a finite group is a mathematical group with a finite number of elements.

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Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

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Free group

In mathematics, the free group FS over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st.

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Free product

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties.

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Generating set of a group

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

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Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Map (mathematics)

In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Mathieu group

In the area of modern algebra known as group theory, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by.

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Permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

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Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself).

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Presentation of a group

In mathematics, one method of defining a group is by a presentation.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Restriction (mathematics)

In mathematics, the restriction of a function f is a new function f\vert_A obtained by choosing a smaller domain A for the original function f. The notation f is also used.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

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Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

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Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

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Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

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Undecidable problem

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is known to be impossible to construct a single algorithm that always leads to a correct yes-or-no answer.

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Wreath product

In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product.

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Redirects here:

Residue class-wise affine group, Residue class-wise affine groups.

References

[1] https://en.wikipedia.org/wiki/Residue-class-wise_affine_group

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