Editing Group isomorphism

{{Short description|Bijective group homomorphism}}
{{refimprove|date=June 2015}}
In [[abstract algebra]], a '''group isomorphism''' is a [[Function (mathematics)|function]] between two [[Group (mathematics)|groups]] that sets up a [[bijection]] 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.<ref name="Barnard-2017">{{cite book |last1=Barnard |first1=Tony |last2=Neil |first2=Hugh |name-list-style=amp |date=2017 |title=Discovering Group Theory: A Transition to Advanced Mathematics |location=Boca Ratan |publisher=CRC Press |isbn=9781138030169 |page=94}}</ref>

== Definition and notation==

Given two groups <math>(G, *)</math> and <math>(H, \odot),</math> a ''group isomorphism'' from <math>(G, *)</math> to <math>(H, \odot)</math> is a [[bijection|bijective]] [[group homomorphism]] from <math>G</math> to <math>H.</math> Spelled out, this means that a group isomorphism is a bijective function <math>f : G \to H</math> such that for all <math>u</math> and <math>v</math> in <math>G</math> it holds that
<math display="block">f(u * v) = f(u) \odot f(v).</math>

The two groups <math>(G, *)</math> and <math>(H, \odot)</math> are isomorphic if there exists an isomorphism from one to the other.<ref name="Barnard-2017" /><ref name="Budden-1972">{{cite book |last=Budden |first=F. J. |date=1972 |title=The Fascination of Groups |url=https://vdoc.pub/download/the-fascination-of-groups-4qkp907dmbl0 |format=PDF |location=Cambridge |publisher=Cambridge University Press |isbn=0521080169 |access-date=12 October 2022 |page=142 |via=VDOC.PUB}}</ref> This is written
<math display="block">(G, *) \cong (H, \odot).</math><!-- the Unicode symbol ≅ is not visible with all browsers and browser settings -->

Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes
<math display="block">G \cong H.</math>

Sometimes one can even simply write <math>G = H.</math> 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 [[subgroup]]s of the same group. See also the examples.

Conversely, given a group <math>(G, *),</math> a set <math>H,</math> and a [[bijection]] <math>f : G \to H,</math> we can make <math>H</math> a group <math>(H, \odot)</math> by defining
<math display="block">f(u) \odot f(v) = f(u * v).</math>

If <math>H = G</math> and <math>\odot = *</math> then the bijection is an [[automorphism]] (''q.v.'').

Intuitively, group theorists view two isomorphic groups as follows: For every element <math>g</math> of a group <math>G,</math> there exists an element <math>h</math> of <math>H</math> such that <math>h</math> "behaves in the same way" as <math>g</math> (operates with other elements of the group in the same way as <math>g</math>). For instance, if <math>g</math> [[Generating set of a group|generates]] <math>G,</math> then so does <math>h.</math> This implies, in particular, that <math>G</math> and <math>H</math> are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.

An isomorphism of groups may equivalently be defined as an [[invertible function|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 all [[real number]]s under addition, <math>(\R, +)</math>, is isomorphic to the group of [[positive real numbers]] under multiplication <math>(\R^+, \times)</math>:
*:<math>(\R, +) \cong (\R^+, \times)</math> via the isomorphism <math>f(x) = e^x</math>.
* The group <math>\Z</math> of [[integer]]s (with addition) is a subgroup of <math>\R,</math> and the [[factor group]] <math>\R/\Z</math> is isomorphic to the group <math>S^1</math> of [[complex number]]s of [[absolute value]] 1 (under multiplication):
*:<math>\R/\Z \cong S^1</math>
* The [[Klein four-group]] is isomorphic to the [[Direct product of groups|direct product]] of two copies of <math>\Z_2 = \Z/2\Z</math>, and can therefore be written <math>\Z_2 \times \Z_2.</math> Another notation is <math>\operatorname{Dih}_2,</math> because it is a [[dihedral group]].
* Generalizing this, for all [[parity (mathematics)|odd]] <math>n,</math> <math>\operatorname{Dih}_{2 n}</math> is isomorphic to the direct product of <math>\operatorname{Dih}_n</math> and <math>\Z_2.</math>
* If <math>(G, *)</math> is an [[infinite cyclic group]], then <math>(G, *)</math> 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 <math>(\R, +)</math> is isomorphic to the group <math>(\Complex, +)</math> of all complex numbers under addition.<ref>{{cite journal|last1= Ash|year=1973|title=A Consequence of the Axiom of Choice|journal=Journal of the Australian Mathematical Society|volume=19|issue=3|pages=306–308|doi=10.1017/S1446788700031505|url=http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_19_03%2FS1446788700031505a.pdf&code=d2e5b0d7bbbbe7368eb4aa14d4bda045|access-date=21 September 2013|doi-access=free}}</ref>
* The group <math>(\Complex^*, \cdot)</math> of non-zero complex numbers with multiplication as the operation is isomorphic to the group <math>S^1</math> mentioned above.

==Properties==

The [[kernel (algebra)|kernel]] of an isomorphism from <math>(G, *)</math> to <math>(H, \odot)</math> is always {e<sub>G</sub>}, where e<sub>G</sub> is the [[identity element|identity]] of the group <math>(G, *)</math>

If <math>(G, *)</math> and <math>(H, \odot)</math> are isomorphic, then <math>G</math> is [[Abelian group|abelian]] if and only if <math>H</math> is abelian.

If <math>f</math> is an isomorphism from <math>(G, *)</math> to <math>(H, \odot),</math> then for any <math>a \in G,</math> the [[Order (group theory)|order]] of <math>a</math> equals the order of <math>f(a).</math>

If <math>(G, *)</math> and <math>(H, \odot)</math> are isomorphic, then <math>(G, *)</math> is a [[locally finite group]] if and only if <math>(H, \odot)</math> is locally finite. 

The number of distinct groups (up to isomorphism) of [[order of a group|order]] <math>n</math> is given by [[integer sequence|sequence]] 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 <math>(\Z_n, +_n),</math> where <math>+_n</math> denotes addition [[modular arithmetic|modulo]] <math>n.</math>

Let <math>G</math> be a cyclic group and <math>n</math> be the order of <math>G.</math> Letting <math>x</math> be a generator of <math>G</math>, <math>G</math> is then equal to <math>\langle x \rangle = \left\{e, x, \ldots, x^{n-1}\right\}.</math> 
We will show that
<math display="block">G \cong (\Z_n, +_n).</math>

Define 
<math display="block">\varphi : G \to \Z_n = \{0, 1, \ldots, n - 1\},</math> so that <math>\varphi(x^a) = a.</math> 
Clearly, <math>\varphi</math> is bijective. Then
<math display="block">\varphi(x^a \cdot x^b) = \varphi(x^{a+b}) = a + b = \varphi(x^a) +_n \varphi(x^b),</math> 
which proves that <math>G \cong (\Z_n, +_n).</math>

== Consequences ==

From the definition, it follows that any isomorphism <math>f : G \to H</math> will map the identity element of <math>G</math> to the identity element of <math>H,</math> 
<math display="block">f(e_G) = e_H,</math>
that it will map [[inverse element|inverses]] to inverses,
<math display="block">f(u^{-1}) = f(u)^{-1} \quad \text{ for all } u \in G,</math>
and more generally, <math>n</math>th powers to <math>n</math>th powers,
<math display="block">f(u^n)= f(u)^n \quad \text{ for all } u \in G,</math> 
and that the inverse map <math>f^{-1} : H \to G</math> is also a group isomorphism.

The [[relation (mathematics)|relation]] "being isomorphic" is an [[equivalence relation]]. If <math>f</math> is an isomorphism between two groups <math>G</math> and <math>H,</math> then everything that is true about <math>G</math> that is only related to the group structure can be translated via <math>f</math> into a true ditto statement about <math>H,</math> and vice versa.

== Automorphisms ==<!-- This section is linked from [[Abelian group]] -->

An isomorphism from a group <math>(G, *)</math> to itself is called an [[automorphism]] of the group. Thus it is a bijection <math>f : G \to G</math> such that
<math display="block">f(u) * f(v) = f(u * v).</math>

The [[image (mathematics)|image]] under an automorphism of a [[conjugacy class]] is always a conjugacy class (the same or another).

The [[function composition|composition]] of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group <math>G,</math> denoted by <math>\operatorname{Aut}(G),</math> itself forms a group, the ''[[automorphism group]]'' of <math>G.</math>

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]]s of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to <math>S_3</math> (which itself is isomorphic to <math>\operatorname{Dih}_3</math>).

In <math>\Z_p</math> for a [[prime number]] <math>p,</math> one non-identity element can be replaced by any other, with corresponding changes in the other elements. The automorphism group is isomorphic to <math>\Z_{p-1}</math> For example, for <math>n = 7,</math> multiplying all elements of <math>\Z_7</math> by 3, modulo 7, is an automorphism of order 6 in the automorphism group, because <math>3^6 \equiv 1 \pmod 7,</math> while lower powers do not give 1. Thus this automorphism generates <math>\Z_6.</math> There is one more automorphism with this property: multiplying all elements of <math>\Z_7</math> by 5, modulo 7. Therefore, these two correspond to the elements 1 and 5 of <math>\Z_6,</math> in that order or conversely.

The automorphism group of <math>\Z_6</math> is isomorphic to <math>\Z_2,</math> because only each of the two elements 1 and 5 generate <math>\Z_6,</math> so apart from the identity we can only interchange these.

The automorphism group of <math>\Z_2 \oplus \Z_2 \oplus \Z_2 = \operatorname{Dih}_2 \oplus \Z_2</math> 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 <math>(1,0,0).</math> Any of the remaining 6 can be chosen to play the role of (0,1,0). This determines which element corresponds to <math>(1,1,0).</math> For <math>(0,0,1)</math> we can choose from 4, which determines the rest. Thus we have <math>7 \times 6 \times 4 = 168</math> automorphisms. They correspond to those of the [[Fano plane]], of which the 7 points correspond to the 7 {{nowrap|non-identity}} elements. The lines connecting three points correspond to the group operation: <math>a, b,</math> and <math>c</math> on one line means <math>a + b = c,</math> <math>a + c = b,</math> and <math>b + c = a.</math> See also [[General linear group#Over finite fields|general linear group over finite fields]].

For abelian groups, all non-trivial automorphisms are [[outer automorphism]]s.

Non-abelian groups have a non-trivial [[inner automorphism]] group, and possibly also outer automorphisms.

== See also ==

* [[Group isomorphism problem]]
* {{annotated link|Bijection}}

== References ==

* {{cite book |last=Herstein |first=I. N. |date=1975 |title=Topics in Algebra |edition=2nd |location=New York |publisher=John Wiley & Sons |isbn=0471010901}}

{{reflist}}

[[Category:Group theory]]
[[Category:Morphisms]]