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Let G be a group and $a,b \in G$. It is given that $o(a)$ and $o(b)$ are finite. Can you give an example of a group where $o(ab)$ is infinite?

Martin Sleziak
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Sankha
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  • Presumably $\circ(a)$ refers to the order of $a$, right? – yearning4pi Feb 26 '13 at 13:12
  • That's unusual notation if it's supposed to be the order of an element. Probably what you have seen is $o(a)$, where the $o$ is the first letter of 'order'. – Tara B Feb 26 '13 at 13:20
  • Also related: http://math.stackexchange.com/questions/279199/on-order-of-elements-of-a-infinite-group/279218#279218 – user1729 Feb 26 '13 at 13:52

6 Answers6

33

Probably you have seen this effect: mirrors on two opposite walls of a room.

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Take two parallel hyperplanes in $\mathbb R^n$. Reflection in each of them is an isometry of order 2. But their composition is a translation of infinite order.

GEdgar
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Consider the matrices $$A=\left(\begin{array}[cc] .1 & -1\\ 0 & -1\end{array}\right)\quad\text{and}\quad B=\left(\begin{array}[cc].1&0\\ 0 & -1\end{array}\right).$$ You can check that each have order $2$, but their product gives $$(AB)^n=\left(\begin{array}[cc] .1 & n\\ 0 & 1\end{array}\right),$$a matrix without finite order.

Clayton
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    It is interesting to note that if you consider $A,B$ as elements of $\text{PSL}_2(\mathbb{Z})$, and consider them with their normal action on the upper half plane then your answer is really the same as Andreas Caranti's. – Alex Youcis Feb 26 '13 at 13:58
  • See please http://math.stackexchange.com/a/316041/8581. I made another. ;-) and +10 for you. – Mikasa Feb 27 '13 at 17:29
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The standard example is the infinite dihedral group.

Consider the group of maps on $\mathbf{Z}$ $$ D_{\infty} = \{ x \mapsto \pm x + b : b \in \mathbf{Z} \}. $$ Consider the maps $$ \sigma: x \mapsto -x, \qquad \tau: x \mapsto -x + 1, $$ both of order $2$. Their composition $$ \tau \circ \sigma (x) = \tau(\sigma(x)) : x \mapsto x + 1 $$ has infinite order.

Andreas Caranti
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If $\circ(a)$ is the order of $a$, usually denoted $|a|$, looking at permutations of $\mathbb{N}$ are an easy way to find an example.

For example consider the permutations:

$$ a = (12)(34)(56)...$$ $$ b = (23)(45)(67)...$$

It isn't hard to see that both $a$ and $b$ have order 2, but consider the image of $1$ under $(ab)^n$ to see that the order of $ab$ is infinite.

muzzlator
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4

If there's any example at all, then

$$ G = \langle a,b \mid a^m , b^n \rangle $$

is an example for some positive integers $m,n$. I assert that every choice with $m,n > 1$ is an example.

  • In addition: it can be proved using normal form for free products since $G= \langle a \mid a^m \rangle \ast \langle b \mid b^n \rangle \simeq \mathbb{Z}_m \ast \mathbb{Z}_n$. – Seirios Feb 26 '13 at 14:35
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Every direct plane rotation (of angle $2\theta$) is the composite of two orthogonal reflections with fixed lines making an angle of $\theta$. If $2\theta/2\pi$ is irrational, then the rotation of angle $2\theta$ has infinite order.

Olivier Bégassat
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