# solution verification – Basic proof Group theory

Let F be the set of bijective functions 𝑓 ∶ 𝐴 → 𝐴. Is 𝐹 under the composition operations
◦ form a Group (structure similar to ℝ with +)?

Let $$F=lbrace f:Ato A mid f, text{is a bijection}rbrace$$ let us consider the operation $$circ: Atimes A to A$$ the usual composition of functions, we should prove that $$(F,circ)$$ is a group.

$$i)$$ There are the identity function $$id:Ato A$$ which is bijective such that
$$idcirc f=f circ id=f$$.

$$ii)$$ Notice that if $$f,g,hin F$$ since $$f,g,h$$ are bijective then
$$(fcirc g)circ h= fcirc (g circ h)$$ and as requierd the composition of bijective functions is also bijective.

$$iii)$$ For each function $$fin F$$ since $$f$$ is bijective there are $$gin F$$ such that $$fcirc g=gcirc f= id_A$$
Is it correct?