Mathematical group formed from the automorphisms of an object
In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X.
Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group.
Automorphism groups are studied in a general way in the field of category theory.
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an automorphismgroup is also called a symmetry group. A subgroup of an automorphismgroup is sometimes called a transformation group. Automorphism groups...
mathematics, the outer automorphismgroup of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphismgroup of G and Inn(G) is the...
The set of all automorphisms of an object forms a group, called the automorphismgroup. It is, loosely speaking, the symmetry group of the object. In...
outer automorphisms, which are all conjugate by an inner automorphism). The automorphismgroup of Dn is isomorphic to the holomorph of Z {\displaystyle...
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...
{\displaystyle \mathbb {Z} _{7}} by 3, modulo 7, is an automorphism of order 6 in the automorphismgroup, because 3 6 ≡ 1 ( mod 7 ) , {\displaystyle 3^{6}\equiv...
is the full automorphismgroup of An: Aut(An) ≅ Sn. Conjugation by even elements are inner automorphisms of An while the outer automorphism of An of order...
"field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the...
permutation groups, as well as automorphismgroups of Steiner systems. After the Mathieu groups, no new sporadic groups were found until 1965, when the group J1...
another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphismgroup of the graph...
isomorphism. The set of all automorphisms of a group G, with functional composition as operation, itself forms a group, the automorphismgroup of G. It is denoted...
of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements...
+x_{2n-1}x_{2n}.} The outer automorphismgroup of a split reductive group G over a field k is isomorphic to the automorphismgroup of the root datum of G....
mathematical group theory, the automorphismgroup of a free group is a discrete group of automorphisms of a free group. The quotient by the inner automorphisms is...
of automorphisms of g {\displaystyle {\mathfrak {g}}} are denoted Aut ( g ) {\displaystyle {\text{Aut}}({\mathfrak {g}})} , the automorphismgroup of...
mathematics, a group G is said to be complete if every automorphism of G is inner, and it is centerless; that is, it has a trivial outer automorphismgroup and trivial...
theorem shows that to compute the automorphismgroup of G {\displaystyle G} it suffices to compute the automorphismgroups of the Sylow p {\displaystyle p}...
ασ of the group X(F) Any automorphism π of the Dynkin diagram induces an automorphism απ of the group X(F). The Steinberg and Chevalley groups can be constructed...
certain axioms – an automorphism of a projective space thus defined then being an automorphism f of the set of points and an automorphism g of the set of...
of An; further orthogonal groups 2Dn, from the order 2 automorphism of Dn; the new series 2E6, from the order 2 automorphism of E6; the new series 3D4...
inner automorphismgroups are never non-trivial cyclic). All non-abelian simple groups are quasisimple. The subnormal quasisimple subgroups of a group control...
automorphism group has order 2, with the full automorphismgroup being the group 2F4(2). The Tits group occurs as a maximal subgroup of the Fischer group Fi22...
there exists an automorphism σ of G × G that switches the two factors, i.e. σ(g1, g2) = (g2, g1). For another example, the automorphismgroup of Z × Z is...