Editing Group isomorphism (section)

== 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).