I am learning some group cohomology from Romyar Sharifi's online notes. (Everything I have written here can be found on page 5-6)
Let $G$ be a finite multiplicative group. Consider the group ring $\mathbb{Z}[G]$. The augmentation ideal is defined as follows:
Consider the ring homomorphism $\epsilon:\mathbb{Z}[G]\rightarrow\mathbb{Z}$ $$\epsilon:\sum_{g\in G}a_gg\mapsto\sum_{g\in G}a_g$$
The augmentation ideal is the kernel of $\epsilon$.
$\text{}$ An abelian group $A$ (written additively) is called a $G$-module if there is an action of $G$ on $A$ such that
(1) $1\cdot a=a$ for all $a\in A$.
(2) $g\cdot (a+b)=g\cdot a+g\cdot b$ for all $g\in G$ and $a,b\in A$.
(3) $g\cdot(h\cdot a)=(gh)\cdot a$ for all $g,h\in G$ and $a\in A$
The action of $\mathbb{Z}[G]$ on $A$ is defined as $$(\sum_{g\in G}a_gg)\cdot a=\sum_{g\in G}a_g(g\cdot a)$$
The group of $G$-coinvariants is defined as
$$A_G=\frac{G}{I_GA}$$
$\text{}$ I have doubt about the definition of $I_GA$. Is it defined like this: $$I_GA=\{\sum_{g\in G}(g\cdot a_g-a_g) | a_g\in A \}$$
I have one more doubt: It is mentioned that $A_G$ is the largest quotient of $A$ fixed by $G$. Does $G$ act on $A/I_GA$ (if so, how is the action defined)?