Find the elements of the left coset a^2⟨(a^5)(b^1)⟩ in D11.

Question

Find the elements of the left coset a^2⟨(a^5)(b^1)⟩ in D11. Enter your answer as a comma separated list; make sure each coset representation is of the form (e1,e2) where e1 represents the exponent for a and e2 represents the exponent for b.

Answer 1

Your have probably been introduced to $D_{11}$ as symmetry operations on a regular $11$-gon with $a$ as a rotation, and $b$ as a reflection.

The angle brackets mean the subgroup generated by the element $a^5b$. So the first step is to figure out what its elements are.

If you know any geometry, you should know that any combination of rotations and a reflection is a reflection. So $a^5b$ must be a reflection and hence has order 2. So the only other element of $\langle a^5b\rangle$ is 1.

Now all you have to do is premultiply the two elements by $a^2$ to get the coset $\{a^7b,a^2\}$ or in their notation $\{(7,1),(2,0)\}$

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Alternatively, go down the algebraic route. In that case you need to know that $ba=a^{-1}b$. So you start by squaring $a^5b$ to see what you get: $$a^5ba^5b=a^5(ba)a^4b=a^5(a^{-1}b)a^4b=a^4ba^4b=\dots =b^2=1$$

So the subgroup $\langle a^5b\rangle$ only has two elements: $a%5b$ and 1 and you premultiply by $a^2$ to get the answer.