Chapter 6, Problem 33E

Prove property 1 of Theorem 6.3.
Theorem 6.3 Properties of Isomorphisms Acting on Groups
Suppose that $\varphi$is an isomorphism from a group G onto a group G.

Then
1. ${\varphi }^{-1}$is an isomorphism from $\bar{G}$onto G.

2. G is Abelian if and only if $\bar{G}$is Abelian.

3. G is cyclic if and only if $\bar{G}$is cyclic.

4. If K is a subgroup of G, then $\varphi \left(K\right)\left\{\varphi \left(k\right)|k\in K\right\}$is a subgroupof $\bar{G}$.

5. If $\bar{K}$is a subgroup of $\bar{G}$, then ${\varphi }^{-1}\left(\bar{K}\right)\left\{g\in G|\varphi \left(g\right)\in \bar{K}\right\}$is asubgroup of G.

6. $\varphi \left(Z\left(G\right)\right)=Z\left(\bar{K}\right)$.