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My goal is to prove the following about the dihedral group $D_{2n}$:

Prove that every element in $D_{2n}$ has a unique factorization of the form $a^{i}b^{j}$, where $0 \leq i < n$ and $j=0$ or $1.$

I know that the cyclic subgroup $\left \langle a \right \rangle$ has order $n.$ From this, I know that this has index $2.$ Thus $D_{2n}$ is the disjoint union $$\left \langle a \right \rangle \cup \left \langle a \right \rangle b.$$

After this, I am pretty stuck. Am I headed in the right direction? What would be the correct way to finish this proof?

Thanks in advance!

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    $\begingroup$ If you already know that $D_{2n}$ is a disjoint union of $\langle a\rangle$ and $\langle a\rangle b$, then why don't you already know that every element can be written as $a^ib^j$ with $j=0$ or $1$, $0\leq i\lt n$? And once you know there is at least one such factorization, what is the difficulty in proving uniqueness? $\endgroup$ – Arturo Magidin Mar 7 at 22:44
  • $\begingroup$ So, I gather from what you are saying that the proof is almost complete. I just need to show it is unique? $\endgroup$ – MathIsLife12 Mar 7 at 22:57
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    $\begingroup$ Well, you also need to realize why you've already proven the existence of the factorization... $\endgroup$ – Arturo Magidin Mar 7 at 22:58
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You want products of the form:

$a^{i}b^{j}$, where $0 \leq i < n$ and $j=0$ or $1.$

there are $2n$ of these: $a^0,a^1,\ldots,a^{n-1}$ and $a^0b,a^1b,\ldots,a^{n-1}b$. These $2n$ products each give you an element of $D_{2n}$ so as long as they all represent distinct elements, the factorization is unique.

The $n$ products of the form $a^i$ are distinct because $|a| = n$. Likewise, the elements $a^ib$ are distinct because $a^{i_1}b = a^{i_2}b$ implies $a^{i_1} = a^{i_2}$ by cancellation, and then $i_1 = i_2$ since $0\leq i \leq n$. Finally, there's no $a^{i_1} b = a^{i_2}$ because that would imply that $b \in \left<a\right>$.

That means your $2n$ products are in 1-to-1 correspondence with the elements of $D_{2n}$; each element of $D_{2n}$ has a unique representative as one of the products.

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