Here is the problem in the text, I have a specific quetsion about it:
WOrk out the left and right cosets of H in G when G=A4 (alternating group that permutates 4 numbers)
H={e,(12)(34),(13)(24),(14)(23)}
and
G=A4 H={e, (123), (132)}
Okay, I know how to find cosets, my question here is whether I need to go through all the work of finding the cosets of each element in G. My idea is that LaGranges theorem tells us that |G|/|H| gives us the number of unique cosets, so I only need to work out the 4 unique cases. Am I correct here?
No, the cosets are the easy part lol, I just wanted to make sure I was correct in my assumption that I only need to work out 2 cases (3 possible variations, and 1 of them is my subgroup H). I was just validating that I can be lazy rather than work out all 12 possible cosets
Haha, what is the point of theorems and corrolarys if you can't be a little lazy?
But think about it. We know that the relation which describes the concept of a coset (specifically for left and for right) is an equivalence relation. So it partitions . So you'll know your done finding cosets when they've exhausted the elements of
Okay, that makes sense. But one other question I have is concerning right cosets. I remember in class the professor said lagrange only tells you the number of unique left cosets, which makes sense because that is how you do the proof: take the left cosets, then find an element in G - (the union of cosets you've taken) and use that in a coset until you've exhausted all options. But what does LaGrange tell us about right cosets, if anything?
An isomorphism? I think not. What binary operation were you supposing of defining on ? But a bijection, yes.
So we can conclude that
To see this another way, you can easily show that is a bijection. Or equivalently that Lagranges theorem is equally applicable to right cosets. From this we can gather that
which equivalently shows that .