Editing Group (mathematics) (section)

=== Group homomorphisms ===
{{Main|Group homomorphism}}
Group homomorphisms{{efn|The word homomorphism derives from [[Ancient Greek|Greek]] ὁμός—the same and [[wikt:μορφή|μορφή]]—structure. See {{harvnb|Schwartzman|1994|p=108}}.}} are functions that respect group structure; they may be used to relate two groups. A ''homomorphism'' from a group <math>(G,\cdot)</math> to a group <math>(H,*)</math> is a function <math>\varphi : G\to H</math> such that
{{Block indent|left=1.6|<math>\varphi(a\cdot b)=\varphi(a)*\varphi(b)</math> for all elements <math>a</math> and <math>b</math> in {{tmath|1= G }}.}}
It would be natural to require also that <math>\varphi</math> respect identities, {{tmath|1= \varphi(1_G)=1_H }}, and inverses, <math>\varphi(a^{-1})=\varphi(a)^{-1}</math> for all <math>a</math> in {{tmath|1= G }}. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.{{sfn|Lang|2005|loc=§II.3|p=34}}

The ''identity homomorphism'' of a group <math>G</math> is the homomorphism <math>\iota_G : G\to G</math> that maps each element of <math>G</math> to itself. An ''inverse homomorphism'' of a homomorphism <math>\varphi : G\to H</math> is a homomorphism <math>\psi : H\to G</math> such that <math>\psi\circ\varphi=\iota_G</math> and {{tmath|1= \varphi\circ\psi=\iota_H }}, that is, such that <math>\psi\bigl(\varphi(g)\bigr)=g</math> for all <math>g</math> in <math>G</math> and such that <math>\varphi\bigl(\psi(h)\bigr)=h</math> for all <math>h</math> in {{tmath|1= H }}. An ''[[group isomorphism|isomorphism]]'' is a homomorphism that has an inverse homomorphism; equivalently, it is a [[bijective]] homomorphism. Groups <math>G</math> and <math>H</math> are called ''isomorphic'' if there exists an isomorphism {{tmath|1= \varphi : G\to H }}. In this case, <math>H</math> can be obtained from <math>G</math> simply by renaming its elements according to the function {{tmath|1= \varphi }}; then any statement true for <math>G</math> is true for {{tmath|1= H }}, provided that any specific elements mentioned in the statement are also renamed.

The collection of all groups, together with the homomorphisms between them, form a [[category (mathematics)|category]], the [[category of groups]].{{sfn|Mac Lane|1998}}

An [[injective]] homomorphism <math>\phi : G' \to G</math> factors canonically as an isomorphism followed by an inclusion, <math>G' \;\stackrel{\sim}{\to}\; H \hookrightarrow G</math> for some subgroup {{tmath|1= H }} of {{tmath|1= G }}.
Injective homomorphisms are the [[monomorphism]]s in the category of groups.