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 \operatorname{Aut}(X) 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 .Examples
If X is a set with no additional structure, then any bijection from X to itself is an automorphism, and hence the automorphism group of X in this case is precisely the symmetric group of X. If the set X has additional structure, then it may be the case that not all bije
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In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inver
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
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a gr
On étudie des notions de topologie générale: unions et quotients d'espaces topologiques; on approfondit les notions de revêtements et de groupe fondamental,et d'attachements de cellules et on démontre le Théorème de Seifert-van Kampen. Des exemples de surfaces illustrent les techniques de calcul.
We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspensions. We study long exact sequences. We construct Eilenberg-Mac Lane spaces.
For a group G generated by k elements, the Nielsen equivalence classes are defined as orbits of the action of AutF(k), the automorphism group of the free group of rank k, on the set of generating k-tuples of G. Let p >= 3 be prime and G(p) the Gupta-Sidki p-group. We prove that there are infinitely many Nielsen equivalence classes on generating pairs of G(p).