hi all, just an interesting coset problem I'm having trouble with -

Find the right and left cosets in G of the subgroups H and K of G.

; ;

What do you guys think?

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- Oct 25th 2009, 09:44 PMsfspitfire23cosets
hi all, just an interesting coset problem I'm having trouble with -

Find the right and left cosets in G of the subgroups H and K of G.

; ;

What do you guys think? - Oct 26th 2009, 01:56 AMSwlabr
Firstly, you know precisely how many right/left cosets there are for each subgroup, this is just where is your subgroup.

Now, for left cosets we know , and we know that cosets all have equal size (they have size ). Thus, you need to find however many sets of 4 elements not in such that for all 4 elements.

These will be your left cosets. For your right cosets, you apply a similar formula ( ).

Remember that left and right cosets are equal if and only if your subgroup is . I'm afraid I can't tell you off hand whether is normal, although I am pretty positive it is. is definitely normal. So, hopefully, in each case your left and right cosets will be equal. - Oct 26th 2009, 08:30 AMsfspitfire23
Would e*k={e, (123),(132)}?

I think this would be correct...but why is the (132) in there?

thanks - Oct 27th 2009, 12:56 AMSwlabr
This is correct - the coset is just , as when you multiply every element in the subgroup by the identity you get the identity. Similarly, for all . is in the coset because . The brackets mean "the subgroup generated by the set ". Have you come across this notation before?

- Oct 27th 2009, 03:51 PMsfspitfire23
ah indeed! I see now!