Abstract Algebra: Theory and Applications

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

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

3. The quaternions, $Q_8$

4. ${\displaystyle D_4}$

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

6. ${\displaystyle S_4}$

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

8. ${\displaystyle \mathbb{Q}}$

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.

9.

Let $G$ and $H$ be solvable groups. Show that $G\times H$ is also solvable.

10.

If $G$ has a composition (principal) series and if $N$ is a proper normal subgroup of $G\text{,}$ show there exists a composition (principal) series containing $N\text{.}$

11.

Prove or disprove: Let $N$ be a normal subgroup of $G\text{.}$ If $N$ and $G/N$ have composition series, then $G$ must also have a composition series.

12.

Let $N$ be a normal subgroup of $G\text{.}$ If $N$ and $G/N$ are solvable groups, show that $G$ is also a solvable group.

13.

Prove that $G$ is a solvable group if and only if $G$ has a series of subgroups

where $P_i$ is normal in $P_{i+1}$ and the order of $P_{i+1}/P_i$ is prime.

14.

Let $G$ be a solvable group. Prove that any subgroup of $G$ is also solvable.

15.

Let $G$ be a solvable group and $N$ a normal subgroup of $G\text{.}$ Prove that $G/N$ is solvable.

16.

Prove that $D_n$ is solvable for all integers $n\text{.}$

17.

Suppose that $G$ has a composition series. If $N$ is a normal subgroup of $G\text{,}$ show that $N$ and $G/N$ also have composition series.

18.

Let $G$ be a cyclic $p$-group with subgroups $H$ and $K\text{.}$ Prove that either $H$ is contained in $K$ or $K$ is contained in $H\text{.}$

19.

Suppose that $G$ is a solvable group with order $n\ge 2\text{.}$ Show that $G$ contains a normal nontrivial abelian subgroup.

20.

Recall that the commutator subgroup $G^{\prime }$ of a group $G$ is defined as the subgroup of $G$ generated by elements of the form $a^{-1}{b}^{-1}ab$ for $a,b\in G\text{.}$ We can define a series of subgroups of $G$ by $G^{(0)}=G\text{,}$ $G^{(1)}=G^{\prime }\text{,}$ and $G^{(i+1)}=(G^{(i)}{)}^{\prime }\text{.}$

1. Prove that $G^{(i+1)}$ is normal in $(G^{(i)}{)}^{\prime }\text{.}$ The series of subgroups

   $$G^{(0)}=G\supset G^{(1)}\supset G^{(2)}\supset \cdots$$

   is called the derived series of $G\text{.}$

2. Show that $G$ is solvable if and only if $G^{(n)}=\{e\}$ for some integer $n\text{.}$

21.

Suppose that $G$ is a solvable group with order $n\ge 2\text{.}$ Show that $G$ contains a normal nontrivial abelian factor group.

22. Zassenhaus Lemma.

Let $H$ and $K$ be subgroups of a group $G\text{.}$ Suppose also that $H^{\ast }$ and $K^{\ast }$ are normal subgroups of $H$ and $K$ respectively. Then

1. $H^{\ast }(H\cap K^{\ast })$ is a normal subgroup of $H^{\ast }(H\cap K)\text{.}$

2. $K^{\ast }(H^{\ast }\cap K)$ is a normal subgroup of $K^{\ast }(H\cap K)\text{.}$

3. $H^{\ast }(H\cap K)/H^{\ast }(H\cap K^{\ast })\cong K^{\ast }(H\cap K)/K^{\ast }(H^{\ast }\cap K)\cong (H\cap K)/(H^{\ast }\cap K)(H\cap K^{\ast })\text{.}$

23. Schreier's Theorem.

Use the Zassenhaus Lemma to prove that two subnormal (normal) series of a group $G$ have isomorphic refinements.

24.

Use Schreier's Theorem to prove the Jordan-Hölder Theorem.