Cayley graph in the context of "Quaternion"

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

In mathematics, especially group theory, two elements ${\displaystyle a}$ and ${\displaystyle b}$ of a group are conjugate if there is an element ${\displaystyle g}$ in the group such that ${\displaystyle b=gag^{-1}.}$ This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under ${\displaystyle b=gag^{-1}}$ for all elements ${\displaystyle g}$ in the group.

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).

Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric.

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 ${\displaystyle \mathrm {S} _{n}}$ defined over a finite set of ${\displaystyle n}$ symbols consists of the permutations that can be performed on the ${\displaystyle n}$ symbols. Since there are ${\displaystyle n!}$ (${\displaystyle n}$ factorial) such permutation operations, the order (number of elements) of the symmetric group ${\displaystyle \mathrm {S} _{n}}$ is ${\displaystyle n!}$.

Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.

In mathematics, the free group F_S over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axioms (e.g. st = suut but s ≠ t for s,t,u ∈ S). The members of S are called generators of F_S, and the number of generators is the rank of the free group.An arbitrary group G is called free if it is isomorphic to F_S for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st = suut).

A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property.