 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-i/7/b/ii/solution

G / A

is a cyclic group, generated by

g A

. For any

x, y \in G

, there are integers

r, s

and

a, b \in A

such that

x = g^{r} a, y = g^{s} b .

Since

a, b

lie in the center of a group, they commute with every element, in particular with each other and with the integer powers of a group element

g

. Therefore

x y = g^{r} a g^{s} b = g^{r + s} ab = g^{s + r} ba = g^{s} b g^{r} a = y x .

x, y

were arbitrary, $G$ is an abelian group. This is the cyclic quotient by the center.

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