# Looking for group definitions

• Jul 3rd 2008, 06:59 PM
topsquark
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
• Jul 3rd 2008, 07:30 PM
ThePerfectHacker
Quote:

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 $n\geq 3$.