Group theory
1. Let $G$ be a finite group and $p$ a prime dividing $|G|$. If $H$ is a $p$-subgroup of $G$, show that $p \mid\left([G: H]-\left[N_{G}(H): H\right]\right) .$
2. Let $G$ be a finite group, and let $p$ be the smallest prime dividing $|G|$.
i) Show that any subgroup of index $p$ is normal.
ii) Let $P$ be a Sylow $p$-subgroup of $G$, and $H \leq Z(P)$ a cyclic subgroup. Show that $H \unlhd G$ iff $H \leq Z(G)$.
3. Find the order of the group with presentation $\left\langle x, y \mid x^{4}=y^{3}=1, x y=y^{2} x^{2}\right\rangle$.
4. Let $G$ be a group with $G / Z(G)$ abelian, and let $m \in \mathbb{N}$ be odd. Prove that $G^{m}:=\left\{x^{m} \mid x \in G\right\}$ is a normal subgroup of $G$.
5. Let $n \in \mathbb{N}, n \neq 2$. Show that no group of order $2^{n} \cdot 3 \cdot 5$ is simple.
Past exam problems
6. (6.4.16) Let $G$ be a group of order $2 m$, with $m$ odd. Show that $G$ has a unique subgroup of order $m$.
7. (6.1.3) Does there exist a group $G$ with a normal subgroup $H$ such that $G / H$ is not isomorphic to any subgroup of $G$ ? What if $G$ is finite? Abelian?
8. i) (6.8.20) If $G$ is abelian, show that $|\operatorname{Aut}(G)|$ is odd iff $|\operatorname{Aut}(G)|=1$.
ii) (6.2.8) For any group $G$, show that $|\operatorname{Aut}(G)|=1$ iff $|G| \leq 2$.
iii) (6.1.9) If $G$ is finite, show that $\operatorname{Aut}(G)$ acts transitively on $G \backslash\{e\}$ iff $G \cong(\mathbb{Z} / p \mathbb{Z})^{n}$ for some $p$ prime, $n \in \mathbb{N}$.
9. (6.8.21) For $n \in \mathbb{N}$, show that there is a unique group of order $n \operatorname{iff} \operatorname{gcd}(n, \phi(n))=1$ (here $\phi$ is the Euler phi function).
10. (6.7.7) Let $F_{n}$ be the free group on $n$ generators. Show that $F_{n} \cong F_{m}$ iff $n=m$. (Remark: however, for any $n, m$, there exist injections $F_{n} \hookrightarrow F_{m}$ and $F_{m} \hookrightarrow F_{n}$.)
1. Let $G$ be a finite group and $p$ a prime dividing $|G|$. If $H$ is a $p$-subgroup of $G$, show that $p \mid\left([G: H]-\left[N_{G}(H): H\right]\right) .$
2. Let $G$ be a finite group, and let $p$ be the smallest prime dividing $|G|$.
i) Show that any subgroup of index $p$ is normal.
ii) Let $P$ be a Sylow $p$-subgroup of $G$, and $H \leq Z(P)$ a cyclic subgroup. Show that $H \unlhd G$ iff $H \leq Z(G)$.
3. Find the order of the group with presentation $\left\langle x, y \mid x^{4}=y^{3}=1, x y=y^{2} x^{2}\right\rangle$.
4. Let $G$ be a group with $G / Z(G)$ abelian, and let $m \in \mathbb{N}$ be odd. Prove that $G^{m}:=\left\{x^{m} \mid x \in G\right\}$ is a normal subgroup of $G$.
5. Let $n \in \mathbb{N}, n \neq 2$. Show that no group of order $2^{n} \cdot 3 \cdot 5$ is simple.
Past exam problems
6. (6.4.16) Let $G$ be a group of order $2 m$, with $m$ odd. Show that $G$ has a unique subgroup of order $m$.
7. (6.1.3) Does there exist a group $G$ with a normal subgroup $H$ such that $G / H$ is not isomorphic to any subgroup of $G$ ? What if $G$ is finite? Abelian?
8. i) (6.8.20) If $G$ is abelian, show that $|\operatorname{Aut}(G)|$ is odd iff $|\operatorname{Aut}(G)|=1$.
ii) (6.2.8) For any group $G$, show that $|\operatorname{Aut}(G)|=1$ iff $|G| \leq 2$.
iii) (6.1.9) If $G$ is finite, show that $\operatorname{Aut}(G)$ acts transitively on $G \backslash\{e\}$ iff $G \cong(\mathbb{Z} / p \mathbb{Z})^{n}$ for some $p$ prime, $n \in \mathbb{N}$.
9. (6.8.21) For $n \in \mathbb{N}$, show that there is a unique group of order $n \operatorname{iff} \operatorname{gcd}(n, \phi(n))=1$ (here $\phi$ is the Euler phi function).
10. (6.7.7) Let $F_{n}$ be the free group on $n$ generators. Show that $F_{n} \cong F_{m}$ iff $n=m$. (Remark: however, for any $n, m$, there exist injections $F_{n} \hookrightarrow F_{m}$ and $F_{m} \hookrightarrow F_{n}$.)