1. ## Cosets and subgroups

Let $\displaystyle G = U_{19}$. Write down the cosets of the subgroup of $\displaystyle G$ generated by $\displaystyle [7]$; by $\displaystyle [12]$; by $\displaystyle [8]$; by $\displaystyle [2]$. Verify Lagrange's theorm for each case.

Previously $\displaystyle G = U_m$ was the group (under multiplication) of units of $\displaystyle Z/mZ$

What are the subgroups and how do I find the cosets of those subgroups?

I really only need help getting started with one then I'm sure I'll be able to do the rest, I'm just not certain what the question is looking for.

2. for example the subgroup produced by $\displaystyle <2>$ is the set of elements $\displaystyle 2*2 mod 19,2*2*2 mod 19, 2*2*2*2 mod 19$ all the way until u get your identity element, which in this group is 1. That set of elements is your subgroup produced by 2. Check that its a subgroup of $\displaystyle U_19$ in this case since your group is finite just check that <2> is closed under multiplication mod 19 (which is a subgroup test for finite groups). So $\displaystyle <2> = {2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1}$ since <2> generates the whole Group, the set of cosets is just <2> itself. For example, take $\displaystyle <6> = {6, 17, 7, 4, 5, 11, 9, 16, 1}$ now to find the cosets of the subgroup find an element in the group that isn't in <6>, take 2, now just multipily all elements of <6> with 2 under mod 19. Notice that there are 9 elements in <6> so there should be 18/9 = 2 cosets exactly such that their union produces the original group.

so 2<6> = {12, 15, 14, 8, 10, 3, 18, 13, 2}

so the first coset is {6, 17, 7, 4, 5, 11, 9, 16, 1}
the second coset is 2<6> = {12, 15, 14, 8, 10, 3, 18, 13, 2}

for the subgroup <6> in U(19)