I'm a little confused by exactly what this question is asking and am thinking I might be over simplifing it.
The question reads "Find all subgrounps of ".
Does this simply mean list
{null, {0}, {1}, ... , {0, 1}, {0, 2}, ..., {0,1,2,3,4,5,6,7,8,9,10, 11}} as all of the subgrounds?
There is no operation defined on the group, so I'm not sure what else to do?
Thanks!
How do I count the subgroups then? If I say that [0], [1], ... , [11] are the subgroups its not correct, because the closure property is not satisfied, i.e., [1] + [1] is not a member of [1] in , and so [1] is not a subgroup. So I would have to expand that group to include [2], but then I run into the same issue of [1] + [2]. Eventually I would include all elements and rebuild the group .
The same argument applies for all elements of the group (except [0]). So does that mean the only proper subgroup is [0]? Or am I thinking of this wrong?
Thanks
So if I'm thinking of this right then the subgroups are
[0] -> {[0]}
[1] ->
[2] -> {[0], [2], [4], [6], [8], [10]}
[3] -> {[0], [3], [6], [9]}
[4] -> {[0], [4], [8]}
[6] -> {[0], [6]}
Any other generator just results in the set , which would be repetitive.
wicked thanks! those results make it much easier to determine the subgroups so for the generators of distinct subgroups are the integers that divide 60.
But for your second result showing the subgroups of subgroups, isn't that backwards? In my list I have <6> <3>. Not the other way around as your result would predict?