What is the meaning of the parentheses in $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$? I am studying homomorphisms is groups and i saw a theorem saying:
For $g$ in a group $G$, the cosets $gH$ and $Hg$ are the same, and collapsed onto the single element $\phi(g)$ by $\phi$. That is, $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$.
I am confused about the parentheses here. I think $\{\phi(g)\}$ denotes the set, formed by the elements of the mapping, but i am not sure what $[\ ]$ represent. Can anyone tell how should i interpret those parentheses? Or explain the theorem in other words?
Thank you

$$Φ^{-1}\left[\{Φ(g)\}\right]=gH=Hg\tag{1}$$
$(1)$ simply denotes the preimage (the inverse mapping) of the image of the "set": $\{\Phi(g)\} \subseteq G/H$ under $\Phi$, assuming in this case we have that $H$ is normal. Since the "object" being mapped by the homomorphism $\Phi$ is technically a set, instead of enclosing the set in parentheses, as we would for an "element", it's using square brackets, but it would mean the same to write $Φ^{-1}\left(\{Φ(g)\}\right)=gH=Hg$
Added: I agree with Ted Shifrin that your text is being excessively pedantic (pedantic: overly concerned with formal rules and trivial points of learning!)
We more typically refer $\Phi (g)$ as an element in $G/H$, and could express the preimage of an element $\Phi(g) \in H/G$ by simply using:  $$Φ^{-1}[Φ(g)]=gH=Hg \;\;\;\text{ or}\;\;\; \Phi^{-1}\left(\Phi(g)\right) = gH = Hg\tag{2}$$
