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Concept
Antihomomorphism
Summary
In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inverse of an antiautomorphism is also an antiautomorphism.
Informally, an antihomomorphism is a map that switches the order of multiplication. Formally, an antihomomorphism between structures and is a homomorphism , where equals as a set, but has its multiplication reversed to that defined on . Denoting the (generally non-commutative) multiplication on by , the multiplication on , denoted by , is defined by . The object is called the opposite object to (respectively, opposite group, opposite algebra, etc.).
This definition is equivalent to that of a homomorphism (reversing the operation before or after applying the map is equivalent). Formally, sending to and acting as the identity on maps is a functor (indeed, an involution).
In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism,
φ(xy) = φ(y)φ(x)
for all x, y in X.
The map that sends x to x−1 is an example of a group antiautomorphism. Another important example is the transpose operation in linear algebra, which takes row vectors to column vectors. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed.
With matrices, an example of an antiautomorphism is given by the transpose map. Since inversion and transposing both give antiautomorphisms, their composition is an automorphism. This involution is often called the contragredient map, and it provides an example of an outer automorphism of the general linear group GL(n, F), where F is a field, except when = 2 and n = 1 or 2, or = 3 and n = 1 (i.e., for the groups GL(1, 2), GL(2, 2), and GL(1, 3)).
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