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If G is a finite group generated by $a,b \in G$, both $a$ and $b$ having order 2, what can we say about the order of another element $xy \in G$?

I was thinking that we can conclude that $xy$ has finite order because it is contained in G, but nothing else can be concluded. Is this correct or am I missing something?

tdashrom
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  • As an illustration of your point: consider the two transpositions $(1,2)$ and $(2,3)$. Their product has order $3$. – lulu Jan 23 '18 at 18:30
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    Another element $x$ or another element $xy$. Does it matter that your element "factors" somehow? – Randall Jan 23 '18 at 18:32

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Consider the dihedral group $D_n$, a rotation may be written as a product of two reflections (each of which has order $2$), thus if we choose a suitable $n$, we can show that the product of two elements of order $2$ can have any order. (Of course, if we require that it is an element in a finite group, its order is finite).

Lukas Heger
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