Concept
Inner automorphism
Related concepts (16)
Automorphism
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 the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space.
Symmetric group
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 defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .
Projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group.
Perfect group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that G(1) = G (the commutator subgroup equals the group), or equivalently one such that Gab = {1} (its abelianization is trivial). The smallest (non-trivial) perfect group is the alternating group A5.
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. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, is abelian if and only if contains the commutator subgroup of . So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.
Isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.