- Group isomorphism
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. 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.**Definition and notation**Given two groups (

`G`, *) and (`H`, $odot$), a "group isomorphism" from (`G`, *) to (`H`, $odot$) is a bijectivegroup homomorphism from`G`to`H`. Spelled out, this means that a group isomorphism is a bijective function $f\; :\; G\; ightarrow\; H$ 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 abijection $f\; :\; G\; ightarrow\; H$, we can make`H`a group (`H`, $odot$) by defining: $f(u)\; odot\; f(v)\; =\; f(u\; *\; v)$.If

`H`=`G`and $odot$ = * then the bijection is an automorphism ("q.v.")Intuitively, group theorists view two isomorphic groups as follows: For every element "g" of a group "G", there exists an element "h" of "G" such that "h" 'behaves in the same way' as "g" (operates with other elements of the group in the same way as "g"). For instance, if "g" generates "G", then so does "h". This implies in particular that "G" and "H" are in bijective correspondence. So the definition of an isomorphism is quite natural.

**Examples*** The group of all

real number s with addition, ($mathbb\{R\}$,+), is isomorphic to the group of all positive real numbers with multiplication ($mathbb\{R\}$^{+},×):: $(mathbb\{R\},\; +)\; cong\; (mathbb\{R\}^+,\; imes)$via the isomorphism: $f(x)\; =\; e^x$(see

exponential function ).* The group $mathbb\{Z\}$ of

integer s (with addition) is asubgroup of $mathbb\{R\}$, and thefactor group $mathbb\{R\}$/$mathbb\{Z\}$ is isomorphic to the group $S^1$ ofcomplex number s ofabsolute 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

Klein four-group is isomorphic to thedirect product of two copies of $mathbb\{Z\}\_2\; =\; mathbb\{Z\}/2mathbb\{Z\}$ (seemodular arithmetic ), and can therefore be written $mathbb\{Z\}\_2\; imes\; mathbb\{Z\}\_2$. Another notation is Dih_{2}, because it is adihedral group .* If ("G", *) is an

infinite cyclic group , then ("G", *) is isomorphic to the integers (with the addition operation). From n algebraic point of view, this means that the set of all integers (with the addition operation) is the 'only' infinite cyclic group.Some groups can be proven to be isomorphic, relying on the

axiom of choice , while it is even theoretically impossible to construct concrete isomorphisms. Examples:

* The group ($mathbb\{R\}$, +) is isomorphic to the group ($mathbb\{C\}$, +) of allcomplex number s with addition.

* The group ($mathbb\{C\}$^{*}, ·) of non-zero complex numbers with multiplication as operation is isomorphic to the group "S"^{1}mentioned above.**Properties*** The kernel of an isomorphism from ("G", *) to ("H", $odot$), is always {e

_{G}} where e_{G}is the identity of the group ("G", *)* If ("G", *) is isomorphic to ("H",$odot$), and if "G" is abelian then so is "H".

* If ("G", *) is a finite group that is isomorphic to ("H", $odot$) [where "f" is the isomorphism] , then if "a" belongs to "G" and has order "n", then so does "f(a)".

* If ("G", *) is a

locally finite group that is isomorphic to ("H", $odot$), then ("H", $odot$) is also locally finite.* The previous examples illustrate that 'group properties' are always preserved by isomorphisms.

**Consequences**From the definition, it follows that any isomorphism $f\; :\; G\; ightarrow\; H$ 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,: $f(u^\{-1\})\; =\; left\; [\; f(u)\; ight]\; ^\{-1\}$and more generally, "n"th powers to "n"th powers,: $f(u^n)=\; left\; [\; f(u)\; ight]\; ^n$for all`u`in`G`,and that the inverse map $f^\{-1\}\; :\; H\; ightarrow\; G$ is also a group isomorphism.The relation "being isomorphic" satisfies all the axioms of an

equivalence relation . 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 anautomorphism of this group. Thus it is a bijection $f\; :\; G\; ightarrow\; G$ 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). 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 . For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to`S`_{3}and Dih_{3}.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 Z_{7}by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because 3^{6}= 1 ( modulo 7 ), while lower powers do not give 1. Thus this automorphism generates Z_{6}. There is one more automorphism with this property: multiplying all elements of Z_{7}by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of Z_{6}, in that order or conversely.The automorphism group of Z

_{6}is isomorphic to Z_{2}, because only each of the two elements 1 and 5 generate Z_{6}, so apart from the identity we can only interchange these.The automorphism group of Z

_{2}× Z_{2}× Z_{2}= Dih_{2}× Z_{2}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 theFano plane , of which the 7 points correspond to the 7 non-identity elements. 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.For Abelian groups all automorphisms except the trivial one are called

outer automorphism s.Non-Abelian groups have a non-trivial

inner automorphism group, and possibly also outer automorphisms.**References**

*Wikimedia Foundation.
2010.*

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