I'm self-studying homological algebra. I have problems on understanding a theorem about $H_0$. First, I don't know where the bottom row of the commutative diagram comes from, see the red line in figure 1. Second, what does $\mathcal{G}A$ mean? Is it $\mathcal{G}[A]$, a group ring produced by $A$?
Also, what does $NA$ mean in figure 2? $N$ even is not a algebraic structure, is it? (These figures are all from Rotman's Advance Modern Algebra Book II)
Figure 1:
Figure 2:
The bottom row comes from the fact (see below) that $\mathcal{G}A$ is a submodule of $A$, so of course you have a short exact sequence $0\to \mathcal{G}A\to A\to A/\mathcal{G}A\to 0$. More generally, whenever $N$ is a submodule of $M$, you have a short exact sequence $0\to N\to M\to M/N\to 0$
$\mathcal{G}$ fits into the short exact sequence $0\to\mathcal{G}\to \mathbb{Z}G\to \mathbb{Z}\to 0$ so it is the kernel of the augmentation map $\mathbb{Z}G\to \mathbb{Z}$. In other words, it is the augmentation ideal, and so if $A$ is a $G$-module, $\mathcal{G}A$ is just the submodule of $A$ defined by this ideal, that is the submodule generated by $\{xa, x\in\mathcal{G},a\in A\}$.
Similarly, $N$ is defined as an element of the group ring (specifically, $N=\displaystyle\sum_{g\in G}g$), and it is central, therefore $x\mapsto Nx, A\to A$ is a $G$-morphism, and its image is a sub-$G$-module : it clearly deserves the name $NA$ (in the same way as you would write $f(A)$ or $fA$ for the image of a morphism $f$)