Group isomorphism - Citizendia

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

1 Definition and notation
2 Examples
3 Consequences
4 Automorphisms
5 References

Definition and notation

Given two groups (G, *) and (H, $\odot$ ), a group isomorphism from (G, *) to (H, $\odot$ ) is a bijective group homomorphism from G to H. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function Spelled out, this means that a group isomorphism is a bijective function $f : G \rightarrow H$ such that for all u and v in G it holds that

$f(u * v) = f(u) \odot f(v)$ .

The two groups (G, *) and (H, $\odot$ ) are isomorphic if an isomorphism exists. This is written:

$(G, *) \cong (H, \odot)$

Often shorter and simpler notations can be used. Often there is no ambiguity about the group operation, and it can be omitted:

$G \cong H$

Sometimes one can even simply write G = H. Whether such a notation is possible without confusion or ambiguity depends on context. For example, the equals sign is not very suitable when the groups are both subgroups of the same group. See also the examples.

Conversely, given a group (G, *), a set H, and a bijection $f : G \rightarrow H$ , we can make H a group (H, $\odot$ ) by defining

$f(u) \odot f(v) = f(u * v)$ . In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

If H = G and $\odot$ = * then the bijection is an automorphism (q. v. )

Examples

The group of all real numbers with addition, ( $\mathbb{R}$ ,+), is isomorphic to the group of all positive real numbers with multiplication ( $\mathbb{R}$ ⁺,×):

$(\mathbb{R}, +) \cong (\mathbb{R}^+, \times)$

via the isomorphism

f (x) = e x

(see exponential function). In Mathematics, the real numbers may be described informally in several different ways The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x)

The group $\mathbb{Z}$ of integers (with addition) is a subgroup of $\mathbb{R}$ , and the factor group $\mathbb{R}$ / $\mathbb{Z}$ is isomorphic to the group $S 1$ of complex numbers of absolute value 1 (with multiplication):

$\mathbb{R}/\mathbb{Z} \cong S^1$

An isomorphism is given by

$f(x + \mathbb{Z}) = e^{2 \pi xi}$

for every x in $\mathbb{R}$ . The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

The Klein four-group is isomorphic to the direct product of two copies of $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$ (see modular arithmetic), and can therefore be written $\mathbb{Z}_2 \times \mathbb{Z}_2$ . In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Another notation is Dih₂, because it is a dihedral group. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections

Some groups can be proven to be isomorphic, relying on the axiom of choice, while it is even theoretically impossible to construct concrete isomorphisms. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. Examples:

The group ( $\mathbb{R}$ , +) is isomorphic to the group ( $\mathbb{C}$ , +) of all complex numbers with addition. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
The group ( $\mathbb{C}$ ^*, ·) of non-zero complex numbers with multiplication as operation is isomorphic to the group S¹ mentioned above.

Consequences

From the definition, it follows that any isomorphism $f : G \rightarrow H$ will map the identity element of G to the identity element of H,

f (e G) = e H

that it will map inverses to inverses,

$f(u^{-1}) = \left[ f(u) \right]^{-1}$

and more generally, nth powers to nth powers,

$f(u^n)= \left[ f(u) \right]^n$

for all u in G, and that the inverse map $f^{-1} : H \rightarrow G$ is also a group isomorphism.

The relation "being isomorphic" satisfies all the axioms of an equivalence relation. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" If f is an isomorphism between two groups G and H, then everything that is true about G that is only related to the group structure can be translated via f into a true ditto statement about H, and vice versa.

Automorphisms

An isomorphism from a group (G,*) to itself is called an automorphism of this group. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Thus it is a bijection $f : G \rightarrow G$ such that

f (u) * f (v) = f (u * v)

An automorphism always maps the identity to itself. The image under an automorphism of a conjugacy class is always a conjugacy class (the same or another). In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class The image of an element has the same order as that element.

The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group G, denoted by Aut(G), forms itself a group, the automorphism group of G.

For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses. However, in groups where all elements are equal to their inverse this is the trivial automorphism, e. g. in the Klein four-group. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to S₃ and Dih₃.

In Z_p for a prime number p, one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to Z_{p − 1}. For example, for n = 7, multiplying all elements of Z₇ by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because 3⁶ = 1 ( modulo 7 ), while lower powers do not give 1. Thus this automorphism generates Z₆. There is one more automorphism with this property: multiplying all elements of Z₇ by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of Z₆, in that order or conversely.

The automorphism group of Z₆ is isomorphic to Z₂, because only each of the two elements 1 and 5 generate Z₆, so apart from the identity we can only interchange these.

The automorphism group of Z₂ × Z₂ × Z₂ = Dih₂ × Z₂ has order 168, as can be found as follows. All 7 non-identity elements play the same role, so we can choose which plays the role of (1,0,0). Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which corresponds to (1,1,0). For (0,0,1) we can choose from 4, which determines the rest. Thus we have 7 × 6 × 4 = 168 automorphisms. They correspond to those of the Fano plane, of which the 7 points correspond to the 7 non-identity elements. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each The lines connecting three points correspond to the group operation: a, b, and c on one line means a+b=c, a+c=b, and b+c=a. See also general linear group over finite fields. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

For Abelian groups all automorphisms except the trivial one are called outer automorphisms. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner

Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms. In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G

References

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Contents

Definition and notation

Examples

Consequences

Automorphisms

References