The Order of $ab$ and $ba$ in a Group are the Same

Group Theory Problems and Solutions in Mathematics

Problem 291

Let $G$ be a finite group. Let $a, b$ be elements of $G$.

Prove that the order of $ab$ is equal to the order of $ba$.
(Of course do not assume that $G$ is an abelian group.)
 
LoadingAdd to solve later

Sponsored Links

Proof.

Let $n$ and $m$ be the order of $ab$ and $ba$, respectively. That is,
\[(ab)^n=e, (ba)^m=e,\] where $e$ is the identity element of $G$.

We compute
\begin{align*}
e&=(ab)^n=\underbrace{(ab)\cdot (ab) \cdot (ab) \cdots (ab)}_{n\text{ times}}\\[6pt] &=a\underbrace{\cdot (ba)(ba)\cdot (ba) \cdots (ba)}_{n-1\text{ times}}b\\[6pt] &=a(ba)^{n-1}b.
\end{align*}

From this, we obtain
\[(ba)^{n-1}=a^{-1}b^{-1}=(ba)^{-1},\] and thus we have
\[(ba)^n=e.\] Therefore the order $m$ of $ba$ divides $n$.

Similarly, we see that $n$ divides $m$, and hence $m=n$.
Thus the orders of $ab$ and $ba$ are the same.


LoadingAdd to solve later

Sponsored Links

More from my site

  • The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$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 […]
  • Group of Order $pq$ Has a Normal Sylow Subgroup and SolvableGroup of Order $pq$ Has a Normal Sylow Subgroup and Solvable 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.   Definition/Hint For (a), apply Sylow's theorem. To review Sylow's theorem, […]
  • Use Lagrange’s Theorem to Prove Fermat’s Little TheoremUse 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 […]
  • Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57 Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group. Then determine the number of elements in $G$ of order $3$.   Proof. Observe the prime factorization $57=3\cdot 19$. Let $n_{19}$ be the number of Sylow $19$-subgroups of $G$. By […]
  • Non-Abelian Group of Order $pq$ and its Sylow SubgroupsNon-Abelian Group of Order $pq$ and its Sylow Subgroups 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 $q$.   Hint. Use Sylow's theorem. To review Sylow's theorem, check […]
  • Non-Abelian Simple Group is Equal to its Commutator SubgroupNon-Abelian Simple Group is Equal to its Commutator Subgroup Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.   Definitions/Hint. We first recall relevant definitions. A group is called simple if its normal subgroups are either the trivial subgroup or the group […]
  • If the Order of a Group is Even, then the Number of Elements of Order 2 is OddIf the Order of a Group is Even, then the Number of Elements of Order 2 is Odd Prove that if $G$ is a finite group of even order, then the number of elements of $G$ of order $2$ is odd.   Proof. First observe that for $g\in G$, \[g^2=e \iff g=g^{-1},\] where $e$ is the identity element of $G$. Thus, the identity element $e$ and the […]
  • Normal Subgroup Whose Order is Relatively Prime to Its IndexNormal 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 […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

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

More in Group Theory
Group Theory Problems and Solutions in Mathematics
A Simple Abelian Group if and only if the Order is a Prime Number

Let $G$ be a group. (Do not assume that $G$ is a finite group.) Prove that $G$ is a simple...

Close