# 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.

• I have a ques­tion about gen­eral Ar­bital prac­tice here. A math­e­mat­i­cian will prob­a­bly already know what a group ho­mo­mor­phism is, but they prob­a­bly also don’t need the proofs of the Prop­er­ties, for in­stance, and they don’t need the ex­pla­na­tion of the triv­ial group. Should I have split this up into differ­ent lenses in some way?

• so8res: “I would set up the page as fol­lows:

A group ho­mo­mor­phism is X. Key prop­er­ties of group ho­mo­mor­phisms in­clude:

1. Thing. Im­pli­ca­tions im­pli­ca­tions im­pli­ca­tions. (Proof.)

2. Thing. Im­pli­ca­tions im­pli­ca­tions. (Proof.)

I’d then even­tu­ally add an in­tro lens.”

Hav­ing proofs on child pages makes sense to me too.