An automorphism of the Klein four-group shown as a mapping between two Cayley graphs, a permutation in cycle notation, and a mapping between two Cayley tables.
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.
nontrivial automorphism: negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any...
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the...
an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups...
can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. Constructing the automorphism group is at least as difficult...
up to an element of S6 (an inner automorphism)) yields an outer automorphism S6 → S6. This map is an outer automorphism, since a transposition does not...
itself forms a group, the automorphism group of G . {\displaystyle G.} For all abelian groups there is at least the automorphism that replaces the group...
{\displaystyle \mathbb {R} ^{n}} in the usual way, one has the typical toral automorphism on the quotient. The fundamental group of an n-torus is a free abelian...
subgroup. An IA automorphism is thus an automorphism that sends each coset of the commutator subgroup to itself. The IA automorphisms of a group form...
^{0}({\mathfrak {g}})} is known as the outer automorphism group. It is known that the outer automorphism group for a simple Lie algebra g {\displaystyle...
In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is...
generates a subgroup of the automorphism group of S. If S = Spec k is the spectrum of a finite field, then its automorphism group is the Galois group of...
theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup...
mathematics, a Kolmogorov automorphism, K-automorphism, K-shift or K-system is an invertible, measure-preserving automorphism defined on a standard probability...
automorphism interchanging the two types of reflections (properly, a class of outer automorphisms, which are all conjugate by an inner automorphism)...
the automorphism group of An is the symmetric group Sn, with inner automorphism group An and outer automorphism group Z2; the outer automorphism comes...
In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient...
power automorphism of a group is an automorphism that takes each subgroup of the group to within itself. It is worth noting that the power automorphism of...
quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for...
composition is an automorphism. This involution is often called the contragredient map, and it provides an example of an outer automorphism of the general...
isomorphism from a partially ordered set to itself is called an order automorphism. When an additional algebraic structure is imposed on the posets ( S...
of X is called an automorphism. The set of all automorphisms is a subset of End(X) with a group structure, called the automorphism group of X and denoted...
Iskovskikh–Manin (1971) showed that the birational automorphism group of a smooth quartic 3-fold is equal to its automorphism group, which is finite. In this sense...
D4, there is a single non-trivial automorphism (Out = C2, the cyclic group of order 2), while for D4, the automorphism group is the symmetric group on three...
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping...
that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup...
theory, the automorphism group of a free group is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is the outer...
In mathematics, a Sastry automorphism, is an automorphism of a field of characteristic 2 satisfying some rather complicated conditions related to the problem...
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