Sylow Theorems

Find a Sylow 2-subgroup of $\mathbb{Z}_{12}$.

Find all primes $p$ such that $\mathrm{Syl}_{p}(\mathrm{S}_{5})$ is non-empty.

Let $G$ be a group and $H\leq G$. Prove that $gHg^{-1}\cong H$ for any $g\in G$.

Let $G$ be a group and $g,h\in G$. Prove that $|gh|=|hg|$.

Let $G$ be a group and $S$ be a non-empty subset of $G$. Prove that $\mathrm{N}_{G}(S)$ is a subgroup of $G$.

Let $G$ be a finite group, $p$ be a prime number, and $H$ and $K$ be distinct Sylow $p$-subgroups of $G$. Prove that $H\cong K$.

Let $G$ be a group of order 784. Let $P$ be a Sylow 7-subgroup that is not a normal subgroup of $G$. Find the order of $\mathrm{N}_{G}(P)$.

Show that any group of order $130$ is non-simple.

Show that a group of order 200 has a normal Sylow 5-subgroup.

Show that any Sylow $p$-subgroup of a group of order 33 must be normal.

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. All even perfect numbers are of the form $2^{p-1}\left(2^{p}-1\right)$ where both $p$ and $2^{p}-1$ are primes. Prove that any group with an even perfect number order is not simple.

Let $p$ and $q$ be primes such that $p<q$. Let $G$ be a group of order $pq$.

1. Prove that there is only one subgroup $H$ of $G$ of order $q$. Deduce that $H\lhd G$.

2. Prove also that if $q\not\equiv 1\pmod{p}$ then $G$ is cyclic.

Let $G$ be a finite group, and write the order of $G$ as $p^{k}m$ where $k\geq 0$, $m\geq 1$, and $p$ is a prime such that $p\nmid m$. Let $N\lhd G$ such that $p$ does not divide the index of $N$ in $G$. Prove that $N$ contains all Sylow $p$-subgroups of $G$ and vice versa. That is,

1. prove that any Sylow $p$-subgroup of $N$ is also in $G$; and

2. prove that any Sylow $p$-subgroup of $G$ is also in $N$.

Show that any group of order 3325 is abelian.

Let $G$ be a finite group such that $|G|=p^{k}m$ where $k\geq 1$, $m>1$, and $p$ a prime such that $p\nmid m$. Prove that if $m!<|G|$ then $G$ is non-simple.

Prove that a group of order 30 has a normal subgroup of order 5.

Let $p,q$, and $r$ be distinct primes such that $p<q<r$. Let $G$ be a group of order $pqr$. Prove that $G$ is non-simple.(Hint: show that a normal subgroup of order $p$, $q$, or $r$ must exist.)