# What do the cosets of the group $E/Z(E)$ look like? (E is the quantum error group and Z(E) is the centre of E)

If I have the quantum error group $$E$$ which contains elements of the form:

{$$\pm w_{1} \otimes \dots \otimes w_{n}, \pm i w_{1} \otimes \dots \otimes w_{n}$$}

The centre of $$E$$:

$$Z(E) = =$${ $$\pm I, \pm iI$$ }

The quotient group:

$$E/Z(E) =$$ {$$eZ(E): e \in E$$}

I am finding it difficult to see how the quotient group is an abelian group. However, I think this is because I can't visualist what cosets of the form $$eE(Z)$$ might look like? Would someone be able to give an example of one of these cosets?

The elements of the n-fold Pauli group are either commute or anti-commute. In the quotient over the center they will all commute, since $$-1$$ wouldn't matter. Note that a general element of the Pauli group has the form $$i^k w_{1} \otimes \dots \otimes w_{n},$$ where $$k=0,1,2,3$$ and $$w_j$$ are Pauli matrices. Four elements for each $$k$$ together form a coset, and as a coset representative you can just take $$w_{1} \otimes \dots \otimes w_{n}$$.
• Thank you for your response, I understand most of it now, except for this part: " In the quotient over the center they will all commute, since $−1$ wouldn't matter. " Can you explain why $-1$ doesn't matter here? Oct 13, 2023 at 15:30
• $-1$ is in the center $\{\pm 1,\pm i\}$ which we factor out. More precisely, it's coset representatives which commute up to factors in the center. Oct 13, 2023 at 15:38