From this question/answer, https://math.stackexchange.com/a/2843091/141334, I learned the word that $T$ denotes the $2$-torsion subgroup, i.e., the subgroup comprising elements of order dividing $2.$
What is the $T$ afterall defined? Can one give some example, say $M=\mathbb{Z}_n$ as a cyclice group of order $n$?
Is it correct to say that $T$ denotes the $2$-torsion subgroup of $G$, which means the $$T=H^1(M,\mathbb{Z})/(2 H^1(M,\mathbb{Z}))?$$
How does this $T$ compare to $$ M/(2M)? $$ Say, when $M$ is abelian or not?
For example in this table from https://topospaces.subwiki.org/wiki/Cohomology_of_real_projective_space
question: When $M=\mathbb{Z}_n$ where $n$ is any integer (like $M=\mathbb{Z}_2$) or $M=\mathbb{Z}$, what are the entries of $T$ becomes?
Given a module $M$ over a ring $R$, a $p$-torsion submodule for $p\in R$ is defined as $M[p]=\{m \in M : (\exists n\in \mathbb N_0) , p^nm=0 \}$. The special case where a $M$ is an abelian group, is when you consider consider it as a $\mathbb Z$-module
As for the question when the module is $\mathbb Z_n$ then the $2$-torsion should be:
If $2 \not \mid n$ then $\mathbb Z_n[2]=\{0\}$
and if $2\mid n$ write $n=2^rq$ (where $2 \not \mid q$)
Edit:$$\mathbb Z_n[2]=\{m \in \mathbb Z_n : (\exists k\in \mathbb N_0) , 2^km=0 \}$$ Notice that $(2:q)=1$ therefore using the chinese remainder theorem we have the following: $$\mathbb Z_n = \mathbb Z_{2^r} \oplus \mathbb Z_q$$ and using this decomposition we have $\mathbb Z_n [2] \simeq \mathbb Z_{2^r}$