# To find kernel of $\rho:\mathbb{Z}[G^2] \to \mathbb{Z}[G/H]$, where $\rho((g_1,g_2))=Hg_2-Hg_1$

I am reading Theorem $$2.1$$ of the paper The Volume and Chern-Simons Invariant of a Representation. I ran into a problem in proving the last part of it, i.e., to prove that the cokernel of the map $$F_2 \to F_1$$ is $$K$$.

Here I am trying to describe the problem in a simple form, and I think anyone can understand it without going through the proof of Theorem $$2.1$$.

If $$X$$ is any set, then denote $$\mathbb{Z}[X]$$ to be the free abelian group generated over the set $$X$$. Let $$G$$ be a group and $$H$$ its subgroup and define a group homomorphism $$\rho: \mathbb{Z}[G^2] \to \mathbb{Z}[G/H]$$ as $$\rho(n(g_1,g_2))= nHg_2 -nHg_1.$$ What is the $$\ker(\rho)$$?

As per the requirement of the proof of Theorem $$2.1$$ and if my calculations are correct then the kernel must be generated by the following set $$\{ (hg,g) - (g_2g,g) +(g_1g,g) -(g_1 g_2^{-1}g,g)~|~ g_1,g_2,g \in G, ~h\in H\}.$$

I don't know why $$Hg_2g-Hg_1g-Hg + Hg_1g_2 ^{-1}g=0$$.

Can someone help me in proving this?

• Is H normal in G or is G/H simply the set of cosets?
– user622002
Commented Feb 6, 2020 at 10:01
• Your generating seems weird to me. First $g$ is not on the list of variables on the right, but that is not very important. More to the point, when $m'=0$ and $m=1$, the element is very clearly not in the kernel. Also, what is the point of $h_1$ and $h_2$ when obviously they could be replaced by any $h\in H$, or even just by $1$? Commented Feb 6, 2020 at 10:18
• user622002: $G/H$ is simply the set of cosets. $H$ need not be normal.
– eyp
Commented Feb 6, 2020 at 10:26
• Lama: As per your observations, I have made changes to the question. Moreover, the point that taking $m=1, m'=0$ is right. Let me check my calculations. And I think taking $h=1$ will change the generating set.
– eyp
Commented Feb 6, 2020 at 10:33
• Lama: I have asked a question related to this question at math.stackexchange.com/questions/3536583/….
– eyp
Commented Feb 6, 2020 at 12:04