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.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.