I've spent the day looking at subsets of groups and trying to prove if they are also subgroups of the same group.
For example, take the group U(8) under multiplication modulo 8, you can see very quickly that there is a subgroup isomorphic to U(6) under multiplication modulo 6 (see the image).
I'm wondering if there's a quick way to tell how many subgroups of order n there are of a particular group.
For example, how many subgroups of U(8) are there of orders 1, 2, 3 and 4?
But why?
You might as well be solving Sudoku puzzles.
>>7973969
sylow theorem
>>7974001
Well firstly, for the same reason as one might solve a Sudoku puzzle: fun.
Secondly, (and more importantly) I have a module for which there will be an exam at the end of this year and I want to be able to cut my time down as much as possible on these types of questions.
>>7973969
>multiplication in U(8) mod 8
Nigger what
>>7974011
I don't see where what I've typed is incorrect please correct me. If your having trouble understanding what I mean, I included a handy image which includes U(8) under multiplication modulo 8 on the left and U(6) under multiplication modulo 6 on the right.
>>7974011
Unital multiplication, hence the U.
Also called [math]\left(\mathbb{Z}/n\mathbb{Z}\right)^{\times}[/math]
>>7974028
Terrible notation for obvious reasons.
You should look up what an external direct product is.
>>7974028
U(8) is the group of 8x8 unital matrices.
That's really shitty notation.
