## Problem 510

Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.

Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.

## Problem 461

(a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated.

(b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.

## Problem 460

Let $\Q=(\Q, +)$ be the additive group of rational numbers.

(a) Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.

(b) Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.

## Problem 411

Let $f:R\to R’$ be a ring homomorphism. Let $I’$ be an ideal of $R’$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.

## Problem 322

Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers.

(a) Prove that the map $\exp:\R \to \R^{\times}$ defined by
$\exp(x)=e^x$ is an injective group homomorphism.

(b) Prove that the additive group $\R$ is isomorphic to the multiplicative group
$\R^{+}=\{x \in \R \mid x > 0\}.$

## Problem 283

Let $F$ be a field and let
$H(F)=\left\{\, \begin{bmatrix} 1 & a & b \\ 0 &1 &c \\ 0 & 0 & 1 \end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}$ be the Heisenberg group over $F$.
(The group operation of the Heisenberg group is matrix multiplication.)

Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.

## Problem 163

Let $\Z$ be the additive group of integers. Let $f: \Z \to \Z$ be a group homomorphism.
Then show that there exists an integer $a$ such that
$f(n)=an$ for any integer $n$.