Cyclic group in the context of Finitely generated group

In mathematics, a group action of a group ${\displaystyle G}$ on a set ${\displaystyle S}$ is a group homomorphism from ${\displaystyle G}$ to some group (under function composition) of functions from ${\displaystyle S}$ to itself. It is said that ${\displaystyle G}$ acts on ${\displaystyle S}$.

Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.

In group theory, a subfield of abstract algebra, a cycle graph of a group is an undirected graph that illustrates the various cycles of that group, 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 given group element a, where a, the n-th power of an element a, is defined as the product of a multiplied by itself n times. The element a is said to generate the cycle. In a finite group, some non-zero power of a must be the group identity, which we denote either as e or 1; the lowest such power is the order of the element a, the number of distinct elements in the cycle that it generates. In a cycle graph, the cycle is represented as a polygon, with its vertices representing the group elements and its edges indicating how they are linked together to form the cycle.

In mathematics, an action of a group ${\displaystyle G}$ on a set ${\displaystyle S}$ is, loosely speaking, an operation that takes an element of ${\displaystyle G}$ and an element of ${\displaystyle S}$ and produces another element of ${\displaystyle S.}$ More formally, it is a group homomorphism from ${\displaystyle G}$ to the automorphism group of X (the set of all bijections on ${\displaystyle X}$ along with group operation being function composition). One says that ${\displaystyle G}$ acts on ${\displaystyle S.}$

Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.

In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. It can be described as the symmetry group of a non-square rectangle (with the three non-identity elements being horizontal reflection, vertical reflection and 180-degree rotation), as the group of bitwise exclusive-or operations on two-bit binary values, or more abstractly as ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, the direct product of two copies of the cyclic group of order 2 by the Fundamental Theorem of Finitely Generated Abelian Groups. It was named Vierergruppe (German: [ˈfiːʁɐˌɡʁʊpə] , meaning four-group) by Felix Klein in 1884. It is also called the Klein group, and is often symbolized by the letter ${\displaystyle V}$ or as ${\displaystyle K_{4}}$.

The Klein four-group, with four elements, is the smallest group that is not cyclic. Up to isomorphism, there is only one other group of order four: the cyclic group of order 4. Both groups are abelian.