# Tagged: normal Sylow subgroup

## Problem 566

Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.

## Problem 474

Prove that every finite group of order $72$ is not a simple group.

## Problem 464

Let $G$ be a finite group of order $231=3\cdot 7 \cdot 11$.
Prove that every Sylow $11$-subgroup of $G$ is contained in the center $Z(G)$.

## Problem 293

Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$.

Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.

## Problem 286

Prove that a group of order $20$ is solvable.

## Problem 278

Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.

## Problem 245

Let $p, q$ be prime numbers such that $p>q$.
If a group $G$ has order $pq$, then show the followings.

(a) The group $G$ has a normal Sylow $p$-subgroup.

(b) The group $G$ is solvable.

## Problem 226

Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.
Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.
Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.

## Problem 118

Let $G$ be a finite group of order $18$.

Show that the group $G$ is solvable.

## Problem 100

Determine whether a group $G$ of the following order is simple or not.

(a) $|G|=100$.
(b) $|G|=200$.

## A Group of Order $pqr$ Contains a Normal Subgroup of Order Either $p, q$, or $r$
Let $G$ be a group of order $|G|=pqr$, where $p,q,r$ are prime numbers such that $p<q<r$.
Show that $G$ has a normal subgroup of order either $p,q$ or $r$.