Definition of Exterior Power in Rotman In Rotman's An Introduction to Algebraic Topology, he defines the exterior power as:

If $M$ is an $A$-module and $p\ge0$, then the $p$th exterior power
  of $M$, denoted by $\bigwedge^pM$, is the abelian group with the
  following presentation: 
Generators: $A\times M\times\dots\times M$ ($p$ factors $M$).
Relations: Some list of relations (I can type them up/screenshot them if needed)

He goes on to say: 

If $F$ is the free abelian group with basis $A\times M\times\dots\times M$ and if $S$ is the subgroup of $F$ generated by the relations, then the coset $(a,m_1,\dots,m_p)+S$ is denoted by $am_1\wedge\dots\wedge m_p$. Thus every element of $\bigwedge^pM$ has an expression (not necessarily unique) of the form $\sum_ja_jm_1^j\wedge\dots\wedge m_p^j$ where $a_j\in A$ and $m_i^j\in M$.

I'm a bit confused by how $S$ is different from $\bigwedge^pM$. They seem to be defined the same way. In particular, $F$ seems to just be $\bigwedge^pM$ without the relations, so by adding in the relations (which gives us the exterior power), we should also get $S$. Obviously, this is wrong, but if someone could explain why, that'd be great.
Thanks!
 A: Perhaps read up on group presentations.  In general you get a free group on the generators,  and then you mod out by the normal subgroup generated by the relations.  Looking at some examples would be a good idea. 

Take the dihedral group.  A presentation is $D_n=\langle r,s\mid r^n,s^2,(rs)^2\rangle $.
So, you start with the free group on two generators,  and mod out by the subgroup generated by the three relations.  
What you get is the symmetries of a regular $n$-gon.  (This group isn't abelian.)

Somewhat more simply, consider the cyclic group of order $n$.  It has presentation $\langle a\mid a^n\rangle $.
Here $F=\Bbb Z$ and $S=n\Bbb Z$.
Each element of $F/S$ is a coset.  For instance,  in $\Bbb Z/n\Bbb Z$, each coset is represented by an integer.  To address your question above,  there is a difference between a coset and its representative.  The representatives of the $S$ cosets come from $F$. This seems to be the source of your confusion.  Take $\Bbb Z_3$.  The coset $[2]=\{2+3k\mid k\in\Bbb Z\}$, which is quite distinct from $2$ itself. 
You are working in the category of abelian groups,  so this link is relevant.    
A: $S$ is the submodule (of $F$) generated by the relations while $\bigwedge^p M$ is the quotient module $F/S$.
