Quotient Group G/G = {identity}? I know this is a basic question, but I'm trying to convince myself of Wikipedia's statement. "The quotient group $G / G$ is isomorphic to the trivial group."
I write the definition for left multiplication because left cosets = right cosets. $ G/G = \{g \in G : gG\} $ But how is this isomorphic to the trivial group, $ \{id_G\} $? $gG$ can't be simplified to $id_G$ ?
Thank you.

$G/G=id_G$ is false. It should be $G/G=\{[id_G]\}$ And actually you could use any element, where the [] mean the equivalence class. Saying G/G=all elements in G is also false, it should be that G/G is a group containing only one element, that is a set that contains all elements in G.
If we go to the definition of quotient group $xy^{-1}\in G\forall x,y\in G$, then, as all elements are related to each other and there's only one equivalence class, therefore the group G/G has one element. As it has one element and the identity must be in the group, then that element is the identity, but it's the identity of the group G/G, not the identity of the group G. The isomorphism G/G->{1} is trivial.
