Possible order of $ab$ when orders of $a$ and $b$ are known.

Question

Let $a,b\in G$ be elements of a finite group $G$. We know $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. In dependence of $m$ and $n$ what are the possible values of $\operatorname{ord}(ab)$?

So strictly speaking I'm looking for a function $f:\mathbb N^2\to\mathcal{P}(\mathbb N)$.

Note that the restriction of $G$ being finite is no weakening of the problem. In case $n,m>1$ we could always add $\infty$ to the result of $f(m,n)$.

If the group is not specified I think $ord (ab)$ can be any multiple of $LCM(m, n)$ which divides the order of the group — brick, Sep 02 '15 at 12:06
if a and b commutes then @brick' claim is true, for example f(x)=x and f(x)=1-x then it isnot true — R.N, Sep 02 '15 at 12:14
@brick I think that's only true if the group is abelian (i.e., the group operation is commutative) — layman, Sep 02 '15 at 12:14
I believe $$f(m,n)=\begin{cases}\mathbb N&m=n\\mathbb N\setminus{1}&m\neq n.\end{cases}$$ But I don't know how to show it. — Thomas Andrews, Sep 02 '15 at 12:47

score 16 · Accepted Answer · answered Sep 02 '15 at 12:13

16

Let $a$ and $b$ be elements of a group $G$. If $a$ has order $m$ and $b$ has order $n$, what can we say about the order of $ab$? The following theorem, proved in Milne's lecture notes on group theory, shows that we can say nothing at all.

THEOREM 1.64 For any integers $m,n, r > 1$, there exists a finite group $G$ with elements $a$ and $b$ such that $a$ has order $m$, $b$ has order $n$, and $ab$ has order $r$.

answered Sep 02 '15 at 12:13

Dietrich Burde

140,055

Can you sketch the proof? – principal-ideal-domain Sep 02 '15 at 12:23
1

For the proof see Theorem $1.64$ here. – Dietrich Burde Sep 02 '15 at 12:27
Nice proof and probably also nice book. Thank you very much. – principal-ideal-domain Sep 02 '15 at 14:45
@DietrichBurde Is it possible to find a particular group where two elements have order three but their product has order 5? – Babai Feb 27 '16 at 18:42
1

Yes, Milne gives such a group in his proof, namely $PSL_2(\mathbb{F}_q)$ for a suitable prime power $q$. – Dietrich Burde Feb 27 '16 at 19:14

Possible order of $ab$ when orders of $a$ and $b$ are known.

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