Quotient Mappings for Vector Spaces and for Groups Below are screencaps of notes from an online Abstract Algebra course. The notes are talking about mappings from a vector space to a quotient one.


If I understand the notes correctly, Part 1 is saying that the quotient vector space $V/W$ is a subspace of $V$, for all subspaces $W$ of $V$. Part 2 goes on to say that this isn't true for a general group. 
But the nonexample that Part 2 gives doesn't make sense to me. Let $G = \mathbb{Z}/ 4\mathbb{Z}$, and $H = \mathbb{Z}/2\mathbb{Z}$. $H \triangleleft
 G$, so $G/H$ is a group of order 2. Thus, $G/H \cong H$, so $G/H$ is a subgroup of G. What I am missing? Thanks!
 A: You need a better understanding of what IS is.
First of all, your understanding of part 1 is incorrect. It does not say that $V/W$ is a subspace of $W$. There's a lot missing in those notes, so let me fill in a few details.
Let $\{w_1,...,w_m\}$ be the given basis for the subspace $W$. One then extends that to a basis for $V$ by adding in more basis vectors denoted $\{v_1,...,v_n\}$, and so altogether we get a basis $\{w_1,...,w_m,v_1,...,v_n\}$ for the vector space $V$. What part 1 says is that the quotient map $f : V \to V/W$ takes $\{v_1,...,v_n\}$ to a basis $\{f(v_1),...,f(v_n)\}$ for the quotient space $V/W$.
What Part 1 probably is also trying to say in addition (although again the notes are deficient on this point) is that $f$ restricts to a vector space isomorphism from $\operatorname{Span}\{v_1,...,v_n\}$ to $V/W$. This does indeed give a subspace of $V$ which is isomorphic to $V/W$, namely the subspace $\operatorname{Span}\{v_1,...,v_n\}$. But you may not conclude from this that the quotient space $V/W$ IS a subspace of $V$. 
Two mathematical objects which are isomorphic are not "the same" object.
Regarding part 2, you are again wrong that $G/H$ IS a subgroup of $G$. But your observation can be correctly reworded: $G/H$ is indeed isomorphic to a subgroup of $G$. 
But that's not the real point of the example in part 2. What happens in part 2 is that there does not exist any subgroup $K < G$ such that the restriction to $K$ of the quotient map $G \mapsto G/H$ is an isomorphism from $K$ to $G/H$.
