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Math Help - Looking for group definitions

  1. #1
    Forum Admin topsquark's Avatar
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    Looking for group definitions

    Is there anywhere I can get a (reasonably comprehensive) list of groups and their definitions? For example, there was a recent post about PSL(2, 9) and I've never heard of it, much less been able to even find it on the internet.

    Thanks!
    -Dan
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    Quote Originally Posted by topsquark View Post
    Is there anywhere I can get a (reasonably comprehensive) list of groups and their definitions? For example, there was a recent post about PSL(2, 9) and I've never heard of it, much less been able to even find it on the internet.
    I do not know. I try to list finite groups.

    1) \mathbb{Z}_n. These are cyclic groups. They are the group of \{ [0]_n,[1]_n,...,[n-1]_n\} i.e. the "integers modulo n". Of course, |\mathbb{Z}_n| = n.

    2) S_n. These are permutation groups. A permutation on \{1,2,...,n\} is a function \{1,2,...,n\}\mapsto \{1,2,...,n\} which is a bijection. It is not hard to show |S_n| = n!. This group is not abelian if n\geq 3. More deep properties include that S_n is not solvable. Called "permutation group".

    3) A_n. Any element in S_n can be written as a product of 2-cycles. It can be shown that if a permutation in S_n is a product of even # of 2-cycles then any other such 2-cycle decomposition must have even # of cycles. Therefore, it is well-defined to say a permuation is even if it can be decomposed into an even # of 2-cycles. This group is a subgroup of S_n. Furthermore, it is a normal subgroup of S_n. It is also simple for n\geq 5. Finally |A_n| = \tfrac{n!}{2}. Called "alternating group".

    4) \text{GL}_n(F). Let F be a finite field and n\geq 2. The set of all n\times n invertible matrices with coefficients in F under multiplication will form this group. This is the "general linear group". Sometimes we write \text{GL}(n,q) where q=|F| (and this is perfectly well-defined because finite fields are unique). The number of elements is (q^n-1)(q^n - q)...(q^n-q^{n-1}).

    5) \text{SL}_n(F). Let F be a finite field and n\geq 2. The set of all n\times n matrices with determinant 1 with coefficients in F under multiplication will form this group. This is the "special linear group". Sometimes we write \text{SL}(n,q) where q=|F|. This group is normal subgroup of \text{GL}(n,q). In fact, the mapping \phi: \text{GL}(n,q)\mapsto \mathbb{F}_q^{\times} (here \mathbb{F}_q^{\times} is the multiplicative group of the finite finite) \phi (M) = \det(M) is a group homomorphism. The kernel is exactly \ker (\phi) = \text{SL}(n,q) thus we see by fundamental homomorphism theorem that \text{SL}(n,q) is normal subgroup of \text{GL}(n,q) and furthermore \text{GL}(n,q)/\text{SL}(n,q)\simeq \mathbb{F}_q^{\times}. Thus, |\text{GL}(n,q)| = |\mathbb{F}_q^{\times}| |\text{SL}(n,q)| which means (q^n-1)(q^n - q)...(q^n-q^{n-1}) = (q^n-1)|\text{SL}(n,q)|. And so we have \text{SL}(n,q) = (q^n-q)...(q^n-q^{n-1}).

    6) \text{PGL}_n(F). Or as we seen \text{PGL}(n,q). This \text{GL}(n,q)/\text{Z}(\text{GL}(n,q)) where \text{Z}(\text{GL}(n,q)) is the group center. A property of this "projective general linear group" is that is is simple. Though, I am NOT sure if it simple for any n and any q. I do know it is simple almost always, but I just do not know if there are any exceptions to this rule. And I do not know how many elements it has, I never studied it.

    7) D_n. The symettries of a regular polygon of n vertices. This group has size 2n. It is always a subgroup of S_n. It has size 2n. And it is always non-abelian when n\geq 3.
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