I'm trying to understand disctinct left cosets… to find distinct left cosets you can start by using Lagrange's theorem which says, $|G| / |H|$ = # of left cosets such that $G$ is a group and $H$ is a subgroup of $G$. So for example what would be the distinct left cosets of $\langle 7 \rangle ≤ U_{32}$?
I know $|U_{32}|=16$, but is $|\langle 7 \rangle| =16$? If so, then $|U_{32}|/|\langle 7 \rangle|= 1$. So there would only be 1 distinct left coset. So would it just be, $ \lbrace 7,21,3,17,31,13,27,9,23,5,19,15,29,11,25 \rbrace$ which is just $\langle 7 \rangle$? Is this the correct distinct left cosets of $\langle 7 \rangle ≤ U_{32}$
Best Answer
The order of $7$ modulo $32$ is actually $4$ as opposed to $16$. So, the number of distinct left cosets of $\langle 7 \rangle$ is $4$. A combination of guess and check along with the fact that $a \in aH$ for any subgroup $H$ of some group $G$ will get us the cosets. So, once I see a particular element of the group in a coset, I don’t need to check the coset corresponding to that element.
$$\langle 7 \rangle = \{1, 7, 17, 23\}$$ $$3\langle 7 \rangle = \{3, 5, 19, 21\}$$ $$9\langle 7 \rangle = \{9, 15, 25, 31\}$$ $$11\langle 7 \rangle = \{11, 13, 27, 29\}.$$