# Tagged: Sylow’s theorem

Sylow’s Theorem Problems and Solutions.

Check out the post “Sylow’s Theorem (summary)” for the statement of Sylow’s theorem and various exercise problems about Sylow’s theorem.

The other popular posts in Group Theory are:

## Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$

## 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 $Z(G)$.

## Every Group of Order 20449 is an Abelian Group

## Prove that a Group of Order 217 is Cyclic and Find the Number of Generators

## Problem 458

Let $G$ be a finite group of order $217$.

**(a)** Prove that $G$ is a cyclic group.

**(c)** Determine the number of generators of the group $G$.

## Non-Abelian Group of Order $pq$ and its Sylow Subgroups

## 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 $q$.

Add to solve later## Sylow Subgroups of a Group of Order 33 is Normal Subgroups

## Problem 278

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

Add to solve later## Group of Order $pq$ Has a Normal Sylow Subgroup and Solvable

## 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.

## Subgroup Containing All $p$-Sylow Subgroups of a Group

## Problem 227

Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.

Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.

Then show that $N$ contains all $p$-Sylow subgroups of $G$.

Add to solve later## If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup

## 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 $G$.

## Group of Order 18 is Solvable

## Problem 118

Let $G$ be a finite group of order $18$. Show that the group $G$ is solvable.

Read solution

## If a Subgroup Contains a Sylow Subgroup, then the Normalizer is the Subgroup itself

## Problem 117

Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$.

Let $H$ be a subgroup of $G$ containing the normalizer $N_G(P)$ of $P$ in $G$. Then show that $N_G(H)=H$.

Read solution

## Are Groups of Order 100, 200 Simple?

## Problem 100

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

(a) $|G|=100$.

(b) $|G|=200$.

Read solution

## A Group of Order $pqr$ Contains a Normal Subgroup of Order Either $p, q$, or $r$

## Problem 81

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 $r$.

Read solution

## If the Order is an Even Perfect Number, then a Group is not Simple

## Problem 74

**(a)** Show that if a group $G$ has the following order, then it is not simple.

- $28$
- $496$
- $8128$

**(b) **Show that if the order of a group $G$ is equal to an even * perfect number* then the group is not simple.

## Sylow’s Theorem (Summary)

In this post we review Sylow’s theorem and as an example we solve the following problem.

## Problem 64

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

Read solution

## A Group with a Prime Power Order Elements Has Order a Power of the Prime.

## Problem 17

Let $p$ be a prime number. Suppose that the order of each element of a finite group $G$ is a power of $p$. Then prove that $G$ is a $p$-group. Namely, the order of $G$ is a power of $p$.

Add to solve later