Is SL(2,Z3) the Only Group with 24 Elements?

[ossito_the-diracian]

i have a question about elements in SL(2,Z3), a,b,c,d are intergers and ad-bc=1 or Det [A]=1. i have to write all the matrices of this group and prove that I do have all of them.

i know that only 3 elements exists in Z3 {[0],[1],[2]} with all others just being repeats. i.e. [-3]=0, [[4]=[1].

i can write 24 elements with ad-bc=1,
i.e. [[1,2],[2,2]] which is [1][2]-[2][2]=[2]-[4]=[-2]=[1]

my problem is that i can't quite write WHY i have found all elements and they are no more, i was trying to appraoch i using contradiction but can't get started

 

[matt grime]

There is a well known formula for the formula of order of finite chevalley groups and finite lie groups, of whic SL(2,Z_3) is one.

The first row may be any of 8 non-zero row vectors, ie a and b can be any pair except 0,0. Now, for each pair, one of the entries must be non-zero, you may now insert any of the elements of Z3 in the slot beneath this non-zero one, and this determines what the remaining 4th entry must be. hence counting them there are 8*3=24 elements.

 

[M98Ranger]

SL(2,Z3) is not the only group with 24 elements. There are other groups with 24 elements, such as Dihedral group D12, Quaternion group Q8, and Symmetric group S4. These groups have different structures and properties, but they all have 24 elements.

To prove that you have found all 24 elements in SL(2,Z3), you need to show that every possible combination of a, b, c, and d results in a unique matrix with ad-bc=1. This can be done by considering all the possible values of a, b, c, and d in Z3 and showing that they satisfy the condition ad-bc=1.

For example, consider the matrix [a,b;c,d]=[2,1;0,2]. This satisfies the condition ad-bc=1 since 2*2-0*1=4-0=1. However, [a,b;c,d]=[2,1;1,2] also satisfies the condition since 2*2-1*1=4-1=3 which is equivalent to 1 in Z3. This shows that there are multiple ways to write the same matrix in SL(2,Z3).

To show that you have found all 24 elements, you can also use the fact that SL(2,Z3) is a subgroup of GL(2,Z3) (the general linear group of 2x2 matrices over Z3). Since GL(2,Z3) has 24 elements, and SL(2,Z3) is a subset of GL(2,Z3), it follows that SL(2,Z3) must also have 24 elements.

In summary, to prove that you have found all 24 elements in SL(2,Z3), you need to show that every possible combination of a, b, c, and d results in a unique matrix with ad-bc=1, and that there are no other matrices in SL(2,Z3) that satisfy this condition. This can be done by considering the properties of SL(2,Z3) and its relationship to GL(2,Z3).