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.

_{G} is the identity of the group $(G,\; *)$
If $(G,\; *)$ and $(H,\; \backslash odot)$ are isomorphic, then $G$ is abelian if and only if $H$ is abelian.
If $f$ is an isomorphism from $(G,\; *)$ to $(H,\; \backslash odot),$ then for any $a\; \backslash in\; G,$ the order of $a$ equals the order of $f(a).$
If $(G,\; *)$ and $(H,\; \backslash odot)$ are isomorphic, then $(G,\; *)$ is a locally finite group if and only if $(H,\; \backslash odot)$ is locally finite.
The number of distinct groups (up to isomorphism) of order $n$ is given by

Definition and notation

Given two groups $(G,\; *)$ and $(H,\; \backslash odot),$ a ''group isomorphism'' from $(G,\; *)$ to $(H,\; \backslash odot)$ is a bijective group homomorphism from $G$ to $H.$ Spelled out, this means that a group isomorphism is a bijective function $f\; :\; G\; \backslash to\; H$ such that for all $u$ and $v$ in $G$ it holds that $$f(u\; *\; v)\; =\; f(u)\; \backslash odot\; f(v).$$ The two groups $(G,\; *)$ and $(H,\; \backslash odot)$ are isomorphic if there exists an isomorphism from one to the other. This is written $$(G,\; *)\; \backslash cong\; (H,\; \backslash odot).$$ Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes $$G\; \backslash cong\; H.$$ 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 $f\; :\; G\; \backslash to\; H,$ we can make $H$ a group $(H,\; \backslash odot)$ by defining $$f(u)\; \backslash odot\; f(v)\; =\; f(u\; *\; v).$$ If $H\; =\; G$ and $\backslash odot\; =\; *$ 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 $H$ 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. Thus, the definition of an isomorphism is quite natural. An isomorphism of groups may equivalently be defined as an invertible group homomorphism (the inverse function of a bijective group homomorphism is also a group homomorphism).Examples

In this section some notable examples of isomorphic groups are listed. * The group of allreal number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s under addition, $(\backslash R,\; +)$, is isomorphic to the group of positive real numbers under multiplication $(\backslash R^+,\; \backslash times)$:
*:$(\backslash R,\; +)\; \backslash cong\; (\backslash R^+,\; \backslash times)$ via the isomorphism $f(x)\; =\; e^x$.
* The group $\backslash Z$ of integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

s (with addition) is a subgroup of $\backslash R,$ and the factor group $\backslash R/\backslash Z$ is isomorphic to the group $S^1$ of complex numbers of absolute value 1 (under multiplication):
*:$\backslash R/\backslash Z\; \backslash cong\; S^1$
* The Klein four-group is isomorphic to the direct product of two copies of $\backslash Z\_2\; =\; \backslash Z/2\backslash Z$, and can therefore be written $\backslash Z\_2\; \backslash times\; \backslash Z\_2.$ Another notation is $\backslash operatorname\_2,$ because it is a dihedral group.
* Generalizing this, for all odd $n,$ $\backslash operatorname\_$ is isomorphic to the direct product of $\backslash operatorname\_n$ and $\backslash Z\_2.$
* If $(G,\; *)$ is an infinite cyclic group, then $(G,\; *)$ is isomorphic to the integers (with the addition operation). From an 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, but the proof does not indicate how to construct a concrete isomorphism. Examples:
* The group $(\backslash R,\; +)$ is isomorphic to the group $(\backslash Complex,\; +)$ of all complex numbers under addition.
* The group $(\backslash Complex^*,\; \backslash cdot)$ of non-zero complex numbers with multiplication as the operation is isomorphic to the group $S^1$ mentioned above.
Properties

The kernel of an isomorphism from $(G,\; *)$ to $(H,\; \backslash odot)$ is always , where esequence
In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...

A000001 in the OEIS. The first few numbers are 0, 1, 1, 1 and 2 meaning that 4 is the lowest order with more than one group.
Cyclic groups

All cyclic groups of a given order are isomorphic to $(\backslash Z\_n,\; +\_n),$ where $+\_n$ denotes addition modulo $n.$ Let $G$ be a cyclic group and $n$ be the order of $G.$ Letting $x$ be a generator of $G$, $G$ is then equal to $\backslash langle\; x\; \backslash rangle\; =\; \backslash left\backslash .$ We will show that $$G\; \backslash cong\; (\backslash Z\_n,\; +\_n).$$ Define $$\backslash varphi\; :\; G\; \backslash to\; \backslash Z\_n\; =\; \backslash ,$$ so that $\backslash varphi(x^a)\; =\; a.$ Clearly, $\backslash varphi$ is bijective. Then $$\backslash varphi(x^a\; \backslash cdot\; x^b)\; =\; \backslash varphi(x^)\; =\; a\; +\; b\; =\; \backslash varphi(x^a)\; +\_n\; \backslash varphi(x^b),$$ which proves that $G\; \backslash cong\; (\backslash Z\_n,\; +\_n).$Consequences

From the definition, it follows that any isomorphism $f\; :\; G\; \backslash to\; H$ 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, $$f(u^)\; =\; f(u)^\; \backslash quad\; \backslash text\; u\; \backslash in\; G,$$ and more generally, $n$th powers to $n$th powers, $$f(u^n)=\; f(u)^n\; \backslash quad\; \backslash text\; u\; \backslash in\; G,$$ and that the inverse map $f^\; :\; H\; \backslash to\; G$ is also a group isomorphism. The relation "being isomorphic" satisfies is 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 an automorphism of the group. Thus it is a bijection $f\; :\; G\; \backslash to\; G$ such that $$f(u)\; *\; f(v)\; =\; f(u\; *\; v).$$ Theimage
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

under an automorphism of a conjugacy class is always a conjugacy class (the same or another).
The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group $G,$ denoted by $\backslash operatorname(G),$ 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 inverses this is the trivial automorphism, e.g. in the Klein four-group. For that group all permutation
In mathematics, a permutation of a Set (mathematics), set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers ...

s of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to $S\_3$ (which itself is isomorphic to $\backslash operatorname\_3$).
In $\backslash Z\_p$ for a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...

$p,$ one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to $\backslash Z\_$ For example, for $n\; =\; 7,$ multiplying all elements of $\backslash Z\_7$ by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because $3^6\; \backslash equiv\; 1\; \backslash pmod\; 7,$ while lower powers do not give 1. Thus this automorphism generates $\backslash Z\_6.$ There is one more automorphism with this property: multiplying all elements of $\backslash Z\_7$ by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of $\backslash Z\_6,$ in that order or conversely.
The automorphism group of $\backslash Z\_6$ is isomorphic to $\backslash Z\_2,$ because only each of the two elements 1 and 5 generate $\backslash Z\_6,$ so apart from the identity we can only interchange these.
The automorphism group of $\backslash Z\_2\; \backslash oplus\; \backslash Z\_2\; \backslash oplus\; \backslash oplus\; \backslash Z\_2\; =\; \backslash operatorname\_2\; \backslash oplus\; \backslash Z\_2$ 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 element corresponds to $(1,1,0).$ For $(0,0,1)$ we can choose from 4, which determines the rest. Thus we have $7\; \backslash times\; 6\; \backslash times\; 4\; =\; 168$ automorphisms. They correspond to those of the Fano plane, of which the 7 points correspond to the 7 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 non-trivial automorphisms are outer automorphisms.
Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.
See also

* Group isomorphism problem *References

* {{reflist Group theory Morphisms