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# automorphism

/aw'teuh mawr"fiz euhm/, n. Math.
an isomorphism from a given set to itself. Cf. inner automorphism, outer automorphism.
[1870-75; AUTO-1 + MORPH- + -ISM]

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in mathematics, a correspondence that associates to every element in a set a unique element of the set (perhaps itself) and for which there is a companion correspondence, known as its inverse, such that one followed by the other produces the identity correspondence (i); i.e., the correspondence that associates every element with itself. In symbols, if f is the original correspondence and g is its inverse, then g(f(a)) = i(a) = a = i(a) = f(g(a)) for every a in the set. Furthermore, operations such as addition and multiplication must be preserved; for example, f(a + b) = f(a) + f(b) and f(ab) = f(a)∙f(b) for every a and b in the set.

The collection of all possible automorphisms for a given set A, denoted Aut(A), forms a group, which can be examined to determine various symmetries in the structure of the set A.

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Universalium. 2010.

### Look at other dictionaries:

• Automorphism — Au to*mor phism, n. Automorphic characterization. H. Spenser. [1913 Webster] …   The Collaborative International Dictionary of English

• 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… …   Wikipedia

• automorphism — /aw teuh mawr fiz euhm/, n. Math. an isomorphism from a given set to itself. Cf. inner automorphism, outer automorphism. [1870 75; AUTO 1 + MORPH + ISM] …   Useful english dictionary

• automorphism — noun Etymology: aut + isomorphism Date: 1903 an isomorphism of a set (as a group) with itself …   New Collegiate Dictionary

• automorphism — noun An isomorphism of a mathematical object or system of objects onto itself …   Wiktionary

• Outer automorphism group — 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 the subgroup consisting of inner automorphisms. The outer automorphism group is usually …   Wikipedia

• Inner automorphism — In abstract algebra an inner automorphism is a function which, informally, involves a certain operation being applied, then another one (x) performed, and then the initial operation being reversed. Sometimes this has a net effect ( take off shoes …   Wikipedia

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• Extensible automorphism — In mathematics, an automorphism of a structure is said to be extensible if, for any embedding of that structure inside another structure, the automorphism can be lifted to the bigger structure.In group theory, an extensible automorphism of a… …   Wikipedia

• Class automorphism — In mathematics, in the realm of group 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 of the automorphism group. Some facts: Every inner… …   Wikipedia