 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/9/c/solution

e

. If

z, w

lie in the center, then for every

g \in G

g (z w) g^{- 1} = (g z g^{- 1}) (g w g^{- 1}) = z w, g z^{- 1} g^{- 1} = (g z g^{- 1})^{- 1} = z^{- 1} .

It is therefore closed under the group operation and inverse elements, and inherits associativity, proving it is a subgroup. Each of its elements commutes with every element of

G

, in particular with every other central element. Hence

Z (G) \leq G and z w = w z for all z, w \in Z (G) .

Thus the center is an abelian subgroup.

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