1. ## Coset help

Q: Suppose that $a$ has order 15. Find all of the left cosets of
$< a^{5} >$ in $< a >$.

Proof. First, lets list the elements of $< a >$
$< a >=\{e, a, a^{2}, a^{3} , a^{4} , a^{5} , a^{6} , a^{7} , a^{8} , a^{9} , a^{10} , a^{11} , a^{12} , a^{13} , a^{14}\}$

The cosets would be:
$< a^{5} >, a < a^{5} >, a^{2} < a^{5} >, a^{3} < a^{5} >, a^{4} < a^{5} >$

I don't understand where the choices $e, a, a^{2}, a^{3}, a^{4}$ came from. I see that there will be 5 cosets, since $|||||=\frac{15}{3}=5$, but I don't understand the choices of the cosets.

As I understand it, there will be more cosets, but cosets parition the group into distinct sets. So, once you have found all the cosets whos union is the orginal group, you are done. Correct?

2. Originally Posted by Danneedshelp
Q: Suppose that $a$ has order 15. Find all of the left cosets of
$< a^{5} >$ in $< a >$.

Proof. First, lets list the elements of $< a >$
$< a >=\{e, a, a^{2}, a^{3} , a^{4} , a^{5} , a^{6} , a^{7} , a^{8} , a^{9} , a^{10} , a^{11} , a^{12} , a^{13} , a^{14}\}$

The cosets would be:
$< a^{5} >, a < a^{5} >, a^{2} < a^{5} >, a^{3} < a^{5} >, a^{4} < a^{5} >$

I don't understand where the choices $e, a, a^{2}, a^{3}, a^{4}$ came from. I see that there will be 5 cosets, since $|||||=\frac{15}{3}=5$, but I don't understand the choices of the cosets.

As I understand it, there will be more cosets, but cosets parition the group into distinct sets. So, once you have found all the cosets whos union is the orginal group, you are done. Correct?

"So, once you have found all the cosets whos union is the orginal group, you are done. Correct?" Well, yes...obviously!

The question here is to understand that a subgroup of a group defines an equivalence relation on the group

whose equivalence classes are precisely the left (or right) cosets of the group wrt that sbgp.

Not only that: if $H\leq G\,,\,x,y,\in G\,,\,then\,\,xH=yH\Longleftrightarrow y^{-1}x\in H$ , so you could check easily whether

two of the sets you were given are or not different cosets.

For example, $a=a^3\Longleftrightarrow a^{-3}a\in$ . But $a^{-3}a=a^{-2}=a^{13}\notin \Longrightarrow$ the cosets

are different.

Tonio