I'm trying to understand how cosets work. I came across with the following question:
Check if there is a subgroup of order $2$ of $U_{20}$ (Euler group) and find its cosets.
In the solution they found that $U_{20}$ has three subgroups of order $2$: $\langle 9\rangle$, $\langle 11\rangle$ and $\langle 19\rangle$. Later they took the group $\langle 9\rangle$ and try to find its cosets in $U_{20}$ – they just said that the cosets are: $\langle 9\rangle, \{3,7\},\{11,19\},\{13,17\}$ without explaining why. I'm familiar with the theorem that the left cosets are $gH=\{gh\,:\,h\in H\}$ and right cosets are $Hg=\{hg\,:\,h\in H\}$, but I don't understand how this gives me the solution. I also know that from Lagrange we get:
$$ [U_{20}\,:\,\langle 9\rangle]=\frac{|U_{20}|}{|\langle 9\rangle|}=\frac{8}{2}=4$$
My question is how they understood that the cosets are $\langle 9\rangle, \{3,7\},\{11,19\},\{13,17\}$?
Best Answer
That is not a theorem but the definition of a coset. They probably did not explain it because they just computed the cosets. For example for $g = 3$ we get $$g \cdot \langle 9 \rangle = 3 \cdot \langle 9 \rangle = 3 \cdot \lbrace 9, 1 \rbrace = \lbrace 3 \cdot 9, 3 \cdot 1 \rbrace = \lbrace 7,3 \rbrace.$$ Now just do the computation for the other elements (not being in the cosets so far) and you will see the four cosets.