If $G$ is a trivial group, then the claim is trivial. So assume that $|G|>1$.
Seeking a contradiction, suppose that $|G|=p^nm$ for some $n,m \in \Z$ and $p$ and $m>1$ are relatively prime.
Let $l$ be a prime factor of $m$. Then by Sylow’s theorem, there exists a Sylow $l$-subgroup of $G$.
The order of a nontrivial element of this subgroup is divisible by the prime $l$ and this contradicts that each element has order power of $p$ since $l$ and $p$ are relatively prime.
If we assume Sylow’s theorem, then the proof of this problem is straightforward.
How about proving it more directly (without using Sylow’s theorem)?
Are Groups of Order 100, 200 Simple?
Determine whether a group $G$ of the following order is simple or not.
Use Sylow's theorem and determine the number of $5$-Sylow subgroup of the group $G$.
Check out the post Sylow’s Theorem (summary) for a review of Sylow's […]
Subgroup Containing All $p$-Sylow Subgroups of a Group
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 […]
Finite Group and a Unique Solution of an Equation
Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that
We give two proofs.
Since $m$ and $n$ are relatively prime […]
Sylow’s Theorem (Summary) In this post we review Sylow's theorem and as an example we solve the following problem.
Show that a group of order $200$ has a normal Sylow $5$-subgroup.
Review of Sylow's Theorem
One of the important theorems in group theory is Sylow's theorem.
Sylow's theorem is a […]
Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4
Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.
Use Sylow's theorem.
(See Sylow’s Theorem (Summary) for a review of Sylow's theorem.)
Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]