# Group of Order $pq$ is Either Abelian or the Center is Trivial

## Problem 30

Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers.

Then show that $G$ is either abelian group or the center $Z(G)=1$.

Contents

## Hint.

Use the result of the problem “If the Quotient by the Center is Cyclic, then the Group is Abelian”.

## Proof.

Since the center $Z(G)$ is a (normal) subgroup  of $G$, the order of $Z(G)$ divides the order of $G$ by Lagrange’s theorem.
Thus the order of $Z(G)$ is one of $1,p,q,pq$.

Suppose that $Z(G)\neq 1$.
Then the order of the quotient group $G/Z(G)$ is one of $1,p,q$.
Hence the group $G/Z(G)$ is a cyclic group.

We conclude that $G$ is abelian group by Problem “If the Quotient by the Center is Cyclic, then the Group is Abelian”.

Therefore, either $Z(G)=1$ or $G$ is abelian.

### More from my site

• The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$ Let $p$ be a prime number. Let $G$ be a non-abelian $p$-group. Show that the index of the center of $G$ is divisible by $p^2$. Proof. Suppose the order of the group $G$ is $p^a$, for some $a \in \Z$. Let $Z(G)$ be the center of $G$. Since $Z(G)$ is a subgroup of $G$, the order […]
• If the Quotient by the Center is Cyclic, then the Group is Abelian Let $Z(G)$ be the center of a group $G$. Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian. Steps. Write $G/Z(G)=\langle \bar{g} \rangle$ for some $g \in G$. Any element $x\in G$ can be written as $x=g^a z$ for some $z \in Z(G)$ and $a \in \Z$. Using […]
• Group of Order 18 is Solvable Let $G$ be a finite group of order $18$. Show that the group $G$ is solvable.   Definition Recall that a group $G$ is said to be solvable if $G$ has a subnormal series $\{e\}=G_0 \triangleleft G_1 \triangleleft G_2 \triangleleft \cdots \triangleleft G_n=G$ such […]
• Use Lagrange’s Theorem to Prove Fermat’s Little Theorem Use Lagrange's Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat's Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.   Before the proof, let us recall Lagrange's Theorem. Lagrange's Theorem If $G$ is a […]
• Abelian Normal subgroup, Quotient Group, and Automorphism Group Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$. Let $\Aut(N)$ be the group of automorphisms of $G$. Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime. Then prove that $N$ is contained in the center of […]
• A Group of Order the Square of a Prime is Abelian Suppose the order of a group $G$ is $p^2$, where $p$ is a prime number. Show that (a) the group $G$ is an abelian group, and (b) the group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ without using the fundamental theorem of abelian […]
• Normal Subgroup Whose Order is Relatively Prime to Its Index Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Suppose that the order $n$ of $N$ is relatively prime to the index $|G:N|=m$. (a) Prove that $N=\{a\in G \mid a^n=e\}$. (b) Prove that $N=\{b^m \mid b\in G\}$.   Proof. Note that as $n$ and […]
• No Finite Abelian Group is Divisible A nontrivial abelian group $A$ is called divisible if for each element $a\in A$ and each nonzero integer $k$, there is an element $x \in A$ such that $x^k=a$. (Here the group operation of $A$ is written multiplicatively. In additive notation, the equation is written as $kx=a$.) That […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### Basic Properties of Characteristic Groups

Definition (automorphism). An isomorphism from a group $G$ to itself is called an automorphism of $G$. The set of all...

Close