Question

a group g is isomorphism to one of its proper subgroup then g=z is it true z=integers geoup

Answer 1

No, it is false.

As a counter-example we can think of a group $\mathbb{Z}^2$. It is isomorphic to the group $2\mathbb{Z} \times \mathbb{Z}$, as $\mathbb{Z} \simeq 2\mathbb{Z}$ which is a proper subgroup. At the same time $\mathbb{Z}^2 \nsim \mathbb{Z}$, as $\mathbb{Z}$ has one generator and $\mathbb{Z}^2$ has two of them.