Troubles to understanding notation and some terminology on cohomology of finite groups I am reading Adem, Milgram's book "Cohomology of finite groups" and I have some troubles with the notation. In particular, I don't get whether they are referring to the cohomology as a ring or as a group. For instance they denote by $H^{*}(G,M)$ for a $G$-module M, the cohomology group of $G$ with (untwisted) local coefficients in M. Whereas, other times they write $H^{*}(G,\mathbb{F}_{p})$ for the cohomology ring of $G$. Also what's the difference between twisted and untwisted coefficients? And why when they are refering to cohomology ring they use only $\mathbb{F}_{p}$ or $\mathbb{F}$ (finite or infinite field) and they don't refer to the cohomology ring for arbitrary $G$-modules?

Cohomology is a ring when you take untwisted coefficients in a ring. (Untwisted means the $G$ action is trivial.) As OmarAntolín-Camarena points out in the comments, you can still get a ring in the twisted case if the coefficients are a $G$-ring.

However, this does explain the terminology in your reference, since group cohomology with untwisted coefficients in a field is a ring, whereas for a general $G$-module, it is not a ring.