Show that a group $G$ in which every element has order 1 or 2 is abelian.

Show that a group of order 6 is either isomorphic to $S_6$ or $\mathbb{Z}/6$.

Show that the group $\mathbb{R}/\mathbb{Z}$ is isomorphic to $(\{z\in \mathbb{C}: |z|=1\}. What is the image of $\mathbb{Q}/\mathbb{Z}$ under the isomorphism found in the previous part?

Prove or disprove: For any natural number $n$ and any group $G$, the subset $H$ of elements of order $n$ in $G$ is a subgroup of $G$.

Prove or disprove: For any natural number $n$ and any abelian group $G$, the subset $H$ of elements of order $n$ in $G$ is a subgroup of $G$.

Prove or disprove: If $\phi:G\to G$ is an epimorphism for a group $G$ then $\phi$ is an isomorphism.

Let $G$ be an ableian group and $x,y\in G$ such that $(|x|,|y|)=1$. Show that $|xy|=|x|\cdot |y|$. Prove that the analogous statement without the abelian hypothesis is false.

List all subgroups of $S_4$. Is $D_8$ a normal subgroup of $S_4$?

Let $G$ be a group, $H\le G$ and $\phi:G\to G$ be a group homomorphism. Suppose $\phi$ restricted to $H$ is an isomorphism from $H$ to $\phi(H)$. Is $\phi$ a monomorphism? Is $\phi$ an epimorphism?

Let $G$ be a group, $A$ and $B$ be normal subgroups of $G$ such that $AB=G$. Show that $G/(A\cap B)\cong G/A\times G/B$.

Let $G$ be a finite group such that 3 does not divide $|G|$ and for all $a,b\in G$, $(ab)^3=a^3b^3$. Show that $G$ is abelian.

Below are some more problems which may have overlap with some homework problems and some problems above.

The exercises below are from the book "Abstract Algebra" by D.S. Dummit and R.M. Foote.

1.1: 1abd, 7, 8 13, 14, 22, 25, 35.

1.2: 1, 4, 7, 18

1.3: 4, 5, 10, 13, 15, 19

1.4: 3, 11

1.6: 3, 4, 6, 9, 18, 25, 26

1.7: 5, 6.

2.1: 4, 6, 7, 10, 15

2.2: 7, 11, 14

2.3: 2, 3, 9, 11, 13, 16, 19, 25

3.1: 3, 5, 12, 16, 22, 33, 36, 37, 39

3.2: 4, 5, 9, 10, 13, 14, 16, 18

3.3: 3, 4, 7

## More Practise Problems

### Matrix groups and solvable groups

Let $F$ be a field. Show the set $B_n(F)$ of upper triangular $n\times n$ matrices with entries in $F$ and with one as the diagonal entries is a group.

Show that $B_2(F)$ is abelian. Compute the commutator subgroup of $B_3(\mathbb{Z}/p)$.

Let $G_0$ be a group. For $i\ge 1$, let $G_i$ be the commutator subgroup of $G_{i-1}$. Show that $G_0$ is abelian iff $G_1$ is the trivial group. Show that $G_0$ is solvable iff $G_n$ is the trivial group for so9me $n$.

Let $G$ be a group and $N$ be a normal subgroup of $G$. Show that $G$ is solvable iff $N$ and $G/N$ are solvable.

### More problems from Dummit and Foote

4.1:1, 4, 8(a), 10

4.2:4, 8, 10, 14

4.3:5, 6, 8, 13, 19, 22, 30, 35

4.4:3, 5, 6, 12, 18

4.5:14, 16, 23, 25, 29(Use Prop 23), 32, 36, 39.

5.5:1, 6, 16, 18.

6.3:1, 2, 4.