I'm reading Rotman's Algebraic Topology and I've come across this definition in the added picture below the line that I don't understand.
From a previous chapter: an abelian group $G$ has generators $B$ and relations $\Delta$ iff $G \approx F/R$, where $F$ is a free abelian group with basis $B$ and $R$ is generated by $\Delta$, where $\Delta$ is a subset of $F$.
How do the relations defined below satisfy the definition defined above? Shouldn't the relations be a subset of some $F$? Why are they defined as a condition that seem to alter elements? What do any of the permutation subscripts mean?
Can someone give me a simple example of a group $C_q(K)$ from the below definition?
For your first question, it is standard practice to write a relation in the form $a=b$ when you mean the element $a-b$ of the free group (so $a=0$ means that $a$ is one of the relations).
For your second question, $\pi 1$ means the permutation $\pi$ applied to the number 1, so $(p_{\pi 0}, p_{\pi 1}, ..., p_{\pi q})$ may be equal to, for example, $(p_4, p_q, ..., p_3)$, if $\pi$ sends 0 to 4, 1 to $q$, $q$ to 3, etc.
For your third question, if $q=1$, then the generators are all pairs of vertices of either of the forms $(p, p)$ or $(p,q)$ if there is an edge from $p$ to $q$. The relations are of the form $(p,p)$ and $(p,q) = -(q,p)$, i.e., $(p,q)+(q,p)$. This is isomorphic to the free abelian group with generators $(p,q)$ where there is an edge from $p$ to $q$ and $p$ is less than $q$ in the ordering on the vertices given by the orientation. This description should hold in general, I think: $C_q(K)$ should be isomorphic to the free abelian group on the elements $\langle p_0, ..., p_q \rangle$ where the $p_i$'s are distinct and span a simplex.