Group homomorphism

A group ho­mo­mor­phism is a func­tion be­tween groups which “re­spects the group struc­ture”.

Definition

For­mally, given two groups \((G, +)\) and \((H, *)\) (which here­after we will ab­bre­vi­ate as \(G\) and \(H\) re­spec­tively), a group ho­mo­mor­phism from \(G\) to \(H\) is a func­tion \(f\) from the un­der­ly­ing set \(G\) to the un­der­ly­ing set \(H\), such that \(f(a) \* f(b) = f(a+b)\) for all \(a, b \in G\).

Examples

  • For any group \(G\), there is a group ho­mo­mor­phism \(1_G: G \to G\), given by \(1_G(g) = g\) for all \(g \in G\). This ho­mo­mor­phism is always bi­jec­tive.

  • For any group \(G\), there is a (unique) group ho­mo­mor­phism into the group \(\{ e \}\) with one el­e­ment and the only pos­si­ble group op­er­a­tion \(e \* e = e\). This ho­mo­mor­phism is given by \(g \mapsto e\) for all \(g \in G\). This ho­mo­mor­phism is usu­ally not in­jec­tive: it is in­jec­tive if and only if \(G\) is the group with one el­e­ment. (Unique­ness is guaran­teed be­cause there is only one func­tion, let alone group ho­mo­mor­phism, from any set \(X\) to a set with one el­e­ment.)

  • For any group \(G\), there is a (unique) group ho­mo­mor­phism from the group with one el­e­ment into \(G\), given by \(e \mapsto e_G\), the iden­tity of \(G\). This ho­mo­mor­phism is usu­ally not sur­jec­tive: it is sur­jec­tive if and only if \(G\) is the group with one el­e­ment. (Unique­ness is guaran­teed this time by the prop­erty proved be­low that the iden­tity gets mapped to the iden­tity.)

  • For any group \((G, +)\), there is a bi­jec­tive group ho­mo­mor­phism to an­other group \(G^{\mathrm{op}}\) given by tak­ing in­verses: \(g \mapsto g^{-1}\). The group \(G^{\mathrm{op}}\) is defined to have un­der­ly­ing set equal to that of \(G\), and group op­er­a­tion \(g +_{\mathrm{op}} h := h + g\).

  • For any pair of groups \(G, H\), there is a ho­mo­mor­phism be­tween \(G\) and \(H\) given by \(g \mapsto e_H\).

  • There is only one ho­mo­mor­phism be­tween the group \(C_2 = \{ e_{C_2}, g \}\) with two el­e­ments and the group \(C_3 = \{e_{C_3}, h, h^2 \}\) with three el­e­ments; it is given by \(e_{C_2} \mapsto e_{C_3}, g \mapsto e_{C_3}\). For ex­am­ple, the func­tion \(f: C_2 \to C_3\) given by \(e_{C_2} \mapsto e_{C_3}, g \mapsto h\) is not a group ho­mo­mor­phism, be­cause if it were, then \(e_{C_3} = f(e_{C_2}) = f(gg) = f(g) f(g) = h h = h^2\), which is not true. (We have used that the iden­tity gets mapped to the iden­tity.)

Properties

  • The iden­tity gets mapped to the iden­tity. (Proof.)

  • The in­verse of the image is the image of the in­verse. (Proof.)

  • The image of a group un­der a ho­mo­mor­phism is an­other group. (Proof.)

  • The com­po­si­tion of two ho­mo­mor­phisms is a ho­mo­mor­phism. (Proof.)

Children:

Parents:

  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.