Exploring Subgroup Inverse Maps in Group Theory

[jimmycricket]

Homework Statement

For a group [itex]G[/itex] consider the map [itex]i:G\rightarrow G , i(g)=g^{-1}[/itex]
For a subgroup [itex]H\subset G[/itex] show that [itex]i(gH)=Hg^{-1}[/itex] and [itex]i(Hg)=g^{-1}H[/itex]

Homework Equations

The Attempt at a Solution

I know that for [itex] g_1,g_2 \in G[/itex] we have [itex]i(g_1g_2)=(g_1g_2)^{-1}=g_2^{-1}g_1^{-1}[/itex]
Then since for any [itex]h\in H, h\in G [/itex] we have [itex]i(g_1h)=(g_1h)^{-1}=h^{-1}g_1^{-1}[/itex]
Is this a good approach to the problem?

 

[pasmith]

Homework Statement

For a group [itex]G[/itex] consider the map [itex]i:G\rightarrow G , i(g)=g^{-1}[/itex]
For a subgroup [itex]H\subset G[/itex] show that [itex]i(gH)=Hg^{-1}[/itex] and [itex]i(Hg)=g^{-1}H[/itex]

Homework Equations

The Attempt at a Solution

I know that for [itex] g_1,g_2 \in G[/itex] we have [itex]i(g_1g_2)=(g_1g_2)^{-1}=g_2^{-1}g_1^{-1}[/itex]
Then since for any [itex]h\in H, h\in G [/itex] we have [itex]i(g_1h)=(g_1h)^{-1}=h^{-1}g_1^{-1}[/itex]
Is this a good approach to the problem?

Working out what [itex]i(gh)[/itex] is for [itex]h \in H[/itex] is certainly a good start.

 

[jimmycricket]

Sorry I should have said I'm actually stuck at this point. Any pointers or hints would be appreciated :)

 

[pasmith]

Sorry I should have said I'm actually stuck at this point. Any pointers or hints would be appreciated :)

You are asked to show that, if [itex]H[/itex] is a subgroup of [itex]G[/itex], then for all [itex]g \in G[/itex], [itex]i(gH) = Hg^{-1}[/itex].

So far you have that if [itex]h \in H[/itex] and [itex]g \in G[/itex] then [itex]i(gh) = h^{-1}g^{-1}[/itex]. You now need to explain why [itex]h^{-1}g^{-1} \in Hg^{-1}[/itex].

 

[jimmycricket]

since [itex]H[/itex] is a subgroup, any [itex]h\in H [/itex] has an inverse element [itex]h^{-1}\in H [/itex] such that [itex]hh^{-1}=h^{-1}h=e[/itex] hence [itex]h^{-1}g^{-1}\in Hg^{-1}[/itex]