$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \setcounter{chapter}{-1}$

## Section11.4Exercises

###### 1

Find all of the abelian groups of order less than or equal to $40$ up to isomorphism.

###### 2

Find all of the abelian groups of order $200$ up to isomorphism.

###### 3

Find all of the abelian groups of order $720$ up to isomorphism.

###### 4

Find all of the composition series for each of the following groups.

1. ${\mathbb Z}_{12}$

2. ${\mathbb Z}_{48}$

3. The quaternions, $Q_8$

4. $D_4$

5. $S_3 \times {\mathbb Z}_4$

6. $S_4$

7. $S_n\text{,}$ $n \geq 5$

8. ${\mathbb Q}$

###### 5

Show that the infinite direct product $G = {\mathbb Z}_2 \times {\mathbb Z}_2 \times \cdots$ is not finitely generated.

###### 6

Let $G$ be an abelian group of order $m\text{.}$ If $n$ divides $m\text{,}$ prove that $G$ has a subgroup of order $n\text{.}$

###### 7

A group $G$ is a torsion group if every element of $G$ has finite order. Prove that a finitely generated abelian torsion group must be finite.

###### 8

Let $G\text{,}$ $H\text{,}$ and $K$ be finitely generated abelian groups. Show that if $G \times H \cong G \times K\text{,}$ then $H \cong K\text{.}$ Give a counterexample to show that this cannot be true in general.