Order of GL(n, Z/p) Group & Sylow p-Subgroups

[e12514]

I need a general formula for the order of the group
GL( n, Z/p ) of invertible nxn matrices with entries in Z/p, under (matrix) multiplication.

I got that,
for n=2 the order is (p^2 -1)(p^2 - p),
for n=3 the order is (p^3 - 1)(p^3 - p)(p^3 - p^2)
...
for n the order is (p^n - 1)(p^n - p)...(p^n - p^(n-1)) (I think?)

which is
p^(n(n-1)/2) * PRODUCT_(i from 1 to n) [p^i - 1]

Is that correct? I'm not sure about it.


Also, the number of Sylow p-subgroups in each case is
n=2 => (p+1) Sylow p-subgroups where the order is p^(2(2-1)/2) = p^1
n=3 => there will be either 1 or (p+1) or (p^2 + p + 1) Sylow p-subgroups where the order is p^3

general case for n => no idea how many... Is it even possible to describe all those Sylow p-subgroups (which will probably be very complicated)? The only thing I know is that they will be of order p^(n(n-1)/2), given my guess was correct.

Can anyone give some help? Thanks.

 

[mathwonk]

duhhh? how many independent sequences of vectors exist? first one chooses any non zero vector, in p^n - 1 ways, then one chooses a vector not collinear with that one in p^n - p ways, then one chooses a vector not coplanar with the first two, in p^n - p^2 ways,... get it?

 

[e12514]

How do you (rigorously) prove it is correct? Just describing it like that is not sufficient, right?

For example, if we use an induction approach, we want to multiply the number of choices by p^n ( p^(n+1) - 1 ) when going from n to (n+1) but I can't explain why the " -1 " constraint is among those " p^(n+1) " choices and not anywhere among the (2n+1) entries of that (n+1)x(n+1) matrix above that nxn matrix, if you know what I mean...
Of course there should be other ways of proving the formula is correct but I don't know how.

And then the number of Sylow p-subgroups - I need even more help on that one.

 

[matt grime]

Why isn't that sufficient? A matrix is in GL(n,F) if and only if its columns are linearly independent.

 

[e12514]

Sorry, what I should have said was, is it "sufficient to prove the formula is correct" just by saying that "it needs to be linearly independent" and "showing the formula was constructed by having (p^n - 1) choices for the first vector, ..., (p^n - p^k) choices for the kth vector" and so on. Is that regarded as a proper proof?

And would you have any ideas as for the number of Sylow p-subgroups in each case? Does there exist a general formula?

 

[matt grime]

The amount of detail you need to insert into a proof depends on the audience. If you're just convincing yourself, or some mathematically minded person, this is fine. If you want to get full marks in an exam, then you may wish to add more details; I wouldn't.

I can't say I've ever thought of the number of Sylows, sorry.

 

[e12514]

Oh right, that's cool. Thanks.

 

[Chris Hillman]

Generally speaking, to obtain full marks it should suffice to convince the grader that you could supply more detail if asked. Generally speaking, the very best students write the minimum required to achieve this standard. Which makes the grader's job much easier

(Forgive me if I guessed incorrectly that this was a homework problem in a modern algebra course, but it is very often set as a problem in such courses. BTW, there is special "Homework help" forum with special rules at PF.)