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?
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.
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?