What's the Order of GL_2(F_5)?

  • Thread starter T-O7
  • Start date
  • Tags
    Group
In summary, the conversation discusses the problem of finding the order of the group GL_n(F_p), the group of all invertible n×n matrices whose elements are from the field F_p=\{0,1,...,p-1\}. The conversation includes a discussion of specific cases, such as GL_2(F_5), and suggests a formula for finding the order of GL_n(F_p) as (p^n - 1)*(p^n - p)*...*(p^n - p^(n - 1)), as well as a method for determining the number of elements in GL_2(F_5). The conversation also mentions the importance of considering linearly independent columns and the difficulty of calculating the order of these groups.
  • #1
T-O7
55
0
Okay, so I need to find the order of the group [tex]GL_2(F_5)[/tex], the group of all invertible 2×2 matrices whose elements are from the field [tex]F_5=\{0,1,2,3,4\}[/tex]. It's a pretty tedious question, but I'm not too confident about my answer. I get a total of 486. Anyone bored enough to help? :redface:
 
Physics news on Phys.org
  • #2
I didn't get 486. How did you do the problem?
 
Last edited:
  • #3
Hurkyl, I think you made a typo.

T-O7, you've over counted somewhere. If you give some info on how you came to 486, we might be able to find what went wrong.
 
  • #4
My method was super-long...
This was how I broke it up:
Case 1: 1 non zero entry - no possible matrices
Case 2: 2 non-zero entries
Possible only if entries are diagonal from each other( i.e.[tex]\left(\begin{array}{cc}0&a\\b&0\end{array}\right)[/tex] or [tex]\left(\begin{array}{cc}a&0\\0&b\end{array}\right)[/tex]), and then [tex]4^2\times2 = 32[/tex] of these are possible.
Case 3: 3 non-zero entries
all combinations are possible, and there are [tex]4^3\times4 = 256[/tex] in total.
Case 4:4 non-zero entries
There are [tex]4^4 = 256[/tex] possible in theory. There are several subcases which must be deleted:
Subcase1: matrices of the form [tex]\left(\begin{array}{cc}a&b\\a&b\end{array}\right)[/tex] and [tex]\left(\begin{array}{cc}a&a\\b&b\end{array}\right)[/tex]. There are [tex]4^2 +4^2 - 4 =28[/tex] of these (overcounting 4 matrices, when a=b)
Subcase2: all other matrices that give a determinant congruent to 0 (mod 5). I found 6 "types" of these matrices, each of which could be rearranged into 2,4, or 8 other combinations to give the same determinant. In total I found [tex]30[/tex] of these (this is where I'm a little skeptical).

This gives me a total of [tex]32 + 256 + (256-28-30) = 486 [/tex] elements. Is there something I missed?
 
Last edited:
  • #5
What about matrices that look like

Code:
/ a a \
\ a b /

?


The typical way of approaching this problem is to consider one row at a time...
 
  • #6
I think those are taken care of in case 4. A matrix of that form won't have determinant congruent to 0 (mod5).
 
  • #7
Hi, your case 4 is a problem. The matrices you find in Subcase1 will have zero determinant mod 5 as well, so why aren't they included in your Subcase2?

Maybe a better way to count your matrices in subcase 4 is to allow anything non-zero for the first row:

[tex]\left(\begin{array}{cc}a&b\\ *&*\end{array}\right)[/tex]

Then ask how many options do you have for the second row to make the rows linearly dependant? Try thinking of the rows as vectors when you answer this question.
 
  • #8
You're right, shmoe, my subcase1 should really be under subcase2, but i separated that case because it was to me the more "trivial" case of 4 non-zero entries having a determinant 0. Subcase2 consists of all other "non-trivial" combinations of the non-zero entries that also give a determinant of zero. Unfortunately, i think i might be missing a couple from subcase2. I'll think about your suggestion!
 
  • #9
Success! :biggrin:
I realized i had neglected 6 matrices in Subcase2, so actually there were 36 matrices in that subcase instead of 30. Therefore, i got [tex]32+256+(256-28-36)=480[/tex], which I think is now right...cuz I need the order to divide 16. :biggrin: I guess you can't be too careful in these type of calculations. Thanks all! :smile:
 
  • #10
T-O7 said:
I'll think about your suggestion!

Do think about it carefully and note it's the same idea as Hurkyl is hinting at, one row at a time, except restricted to the case of all entries non-zero.

With this idea you can actually answer this question in one fell swoop. In fact, you can can come up with a simpe formula for the number of elements in [tex]GL_n(F_q)[/tex] for any prime power q and any n by doing the row by row approach.

Good observation realizing the order had to be divisible by 16 though!
 
  • #11
So it turns out that I'm required to now calculate the order of the general group [tex]GL_n(F_p)[/tex], and after thinking about the linearly independent columns like you guys suggested, it struck me that it actually is surprisingly simple to calculate its order. And yes, I did get a pretty nice formula ([tex](p^n-p^0)(p^n-p^1)...(p^n-p^{n-1}))[/tex], and then just to double check, I plugged in the corresponding numbers for the group I had above, and my original answer of 480 turned out to be the right answer (phew). If only I had thought of this sooner, I wouldn't have had to gone through all of that! But thanks a lot :biggrin:
 
Last edited:
  • #12
lol, this is often how mathematics is done. You do a bunch of tedious calculations, and after that you realize that there is a simpler general way of handling the problem :smile:
 
  • #13
Hi T-O7,

I was actually trying to determine the number of elements in both GL_n(F_p) and SL_n(F_p).

I was looking at your investigation into finding the number of elements in GL_2(F_5). Could you please explain why there are 36 elements discarded in subcase 2 of case 4?
 
  • #14
Can you also explain how you derived the formula (p^n - 1)*(p^n - p)*...*(p^n - p^(n - 1)) please? I get that at first there are p^n-1 options to choose any vector which is non-zero but I'm not sure about the rest. Thanks.
 
  • #15
Hi Shmoe,

I'm also doing a similar mathematics problem at Exeter University. Firstly, could you please explain why the order of the group GL_2(F_5) is divisible by 16? Secondly, could you also explain how the number of elements in the group GL_n(F_p) is ((p^n - 1)*(p^n - p)*...*(p^n - p^(n - 1)) (as mentioned by T-O7) please? Thanks
 

FAQ: What's the Order of GL_2(F_5)?

What is the definition of a group?

A group is a mathematical concept that consists of a set of elements and an operation that combines any two elements to produce a third element within the set. The operation must be associative, have an identity element, and each element must have an inverse.

What is the order of a group?

The order of a group is the number of elements in the set. It represents the size or cardinality of the group.

How do you determine the order of a group?

To determine the order of a group, you can count the number of elements in the set or use the Lagrange's Theorem which states that the order of a subgroup must divide the order of the original group.

What is the significance of the order of a group?

The order of a group is significant because it helps classify and compare groups. It also has implications for the properties and structure of the group.

Can the order of a group be infinite?

Yes, the order of a group can be infinite. This means that the group has an uncountable number of elements in its set.

Similar threads

Back
Top