Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 7 / b / i / Solution

The specified set $A$ is the center of a group $Z(G)$. It contains $e$. If $a,b\in A$ and $y\in G$, thenso $ab\in A$. From $ay=ya$, multiplying on the left and right by $a^{-1}$ gives $y{a}^{-1}=a^{-1}y$, so $a^{-1}\in A$. Hence $A$ is a subgroup.

For $g\in G$ and $a\in A$, the defining commutativity gives $ga{g}^{-1}=ag{g}^{-1}=a$. Thus the image of each element of $A$ under conjugation is in $A$, and indeed $gA{g}^{-1}=A$. Consequently $A=Z(G)$ is a normal subgroup of $G$.