Looking for group definitions

**topsquark** · July 3rd 2008, 05:59 PM

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

**ThePerfectHacker** · July 3rd 2008, 06:30 PM

Originally Posted by topsquark

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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