- #1

- 1,456

- 44

## Homework Statement

Prove that if the groups ##G \cong H## are isomorphic then ##Z(G) \cong Z(H)##

## Homework Equations

## The Attempt at a Solution

Let ##\phi: G \to H## be an isomorphism. Define ##f: Z(G) \to Z(H)## s.t ##f(z) = \phi (z)##.

First, we will show that this map is well-defined, in the sense that elements in ##Z(G)## are always mapped to elements in ##Z(H)## by ##f##:

Let ##z \in Z(G)##. Then ##zg=gz## for all ##g \in G##. This implies that ##f(zg)=f(gz) \implies \phi (z) \phi (g) = \phi (g) \phi (z)## for all ##g##. But ##g## is arbitrary and ##\phi## is bijective, so ##\phi (g)## is an arbitrary element of ##H##. Hence we have ##\phi(z)h=h \phi(z)## for all ##h \in H##, which means ##f(z)h=h f(z)## for all ##h \in H##. So ##f(z) \in Z(H)##.

Next, we show that ##f## is an isomorphism: It is clear that ##f## is a homomorphism, since it is defined in terms of ##\phi##. It remains to show that ##f## is a bijection: ##f## is invertible, and the inverse is ##f^{-1} = \phi^{-1}##.

Hence ##Z(G) \cong Z(H)##