- #1

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## Homework Statement

If H is a subgroup of a group G, and a, b [itex]\in[/itex] G, then the following four conditions are equivalent:

i) [itex] ab^{-1} \in H [/itex]

ii) [itex] a=hb [/itex] for some [itex] h \in H [/itex]

iii) [itex] a \in Hb [/itex]

iv) [itex] Ha=Hb [/itex]

## Homework Equations

cancellation law seems handy, and existence of an inverse, associativity, and other basic things.

## The Attempt at a Solution

[itex] ab^{-1} \in H \Rightarrow \exists h \in H : ab^{-1}=h \Rightarrow (ab^{-1})b=hb \Rightarrow a(bb^{-1})=ae=a=hb \Rightarrow a \in Hb \Rightarrow ...? [/itex]