Free Groups, Generators, and Relations

Let $\mathbb {Q}_{+}$ be the multiplicative group of positive rational numbers.

Prove that the additive group of $\mathbb {Q}$, the rational number field, is not finitely generated.

Note: See also Problems 6.3.1 and 6.6.3.

Let $G$ be the abelian group defined by generators $x$, $y$, and $z$, and relations

Suppose that the group $G$ is generated by elements $x$ and $y$ that satisfy $x^{5}y^{3} = x^{8}y^{5} = 1$. Is $G$ the trivial group?

Let $G$ be a group with generators $a$ and $b$ satisfying

Is $G$ trivial?

Let the group $G$ be generated by two elements, $a$ and $b$, both of order 2. Prove that $G$ has a subgroup of index 2.

Let $G_{n}$ be the free group on $n$ generators. Show that $G_{2}$ and $G_{3}$ are not isomorphic.

Let $G$ be an abelian group which is generated by, at most, $n$ elements. Show that each subgroup of $G$ is again generated by, at most, $n$ elements.

Determine all finitely generated abelian groups $G$ which have only finitely many automorphisms.

Let $A$ be a finite abelian group, and $m$ the maximum of the orders of the elements of $A$. Put $S = \{ a \in A \mid \left|a\right| = m\}$. Prove that $A$ is generated by $S$.