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
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
I do not know. I try to list finite groups.
1)$\displaystyle \mathbb{Z}_n$. These are cyclic groups. They are the group of $\displaystyle \{ [0]_n,[1]_n,...,[n-1]_n\}$ i.e. the "integers modulo $\displaystyle n$". Of course, $\displaystyle |\mathbb{Z}_n| = n$.
2)$\displaystyle S_n$. These are permutation groups. A permutation on $\displaystyle \{1,2,...,n\}$ is a function $\displaystyle \{1,2,...,n\}\mapsto \{1,2,...,n\}$ which is a bijection. It is not hard to show $\displaystyle |S_n| = n!$. This group is not abelian if $\displaystyle n\geq 3$. More deep properties include that $\displaystyle S_n$ is not solvable. Called "permutation group".
3)$\displaystyle A_n$. Any element in $\displaystyle S_n$ can be written as a product of $\displaystyle 2$-cycles. It can be shown that if a permutation in $\displaystyle S_n$ is a product of even # of $\displaystyle 2$-cycles then any other such $\displaystyle 2$-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 $\displaystyle 2$-cycles. This group is a subgroup of $\displaystyle S_n$. Furthermore, it is a normal subgroup of $\displaystyle S_n$. It is also simple for $\displaystyle n\geq 5$. Finally $\displaystyle |A_n| = \tfrac{n!}{2}$. Called "alternating group".
4)$\displaystyle \text{GL}_n(F)$. Let $\displaystyle F$ be a finite field and $\displaystyle n\geq 2$. The set of all $\displaystyle n\times n$ invertible matrices with coefficients in $\displaystyle F$ under multiplication will form this group. This is the "general linear group". Sometimes we write $\displaystyle \text{GL}(n,q)$ where $\displaystyle q=|F|$ (and this is perfectly well-defined because finite fields are unique). The number of elements is $\displaystyle (q^n-1)(q^n - q)...(q^n-q^{n-1})$.
5)$\displaystyle \text{SL}_n(F)$. Let $\displaystyle F$ be a finite field and $\displaystyle n\geq 2$. The set of all $\displaystyle n\times n$ matrices with determinant $\displaystyle 1$ with coefficients in $\displaystyle F$ under multiplication will form this group. This is the "special linear group". Sometimes we write $\displaystyle \text{SL}(n,q)$ where $\displaystyle q=|F|$. This group is normal subgroup of $\displaystyle \text{GL}(n,q)$. In fact, the mapping $\displaystyle \phi: \text{GL}(n,q)\mapsto \mathbb{F}_q^{\times}$ (here $\displaystyle \mathbb{F}_q^{\times}$ is the multiplicative group of the finite finite) $\displaystyle \phi (M) = \det(M)$ is a group homomorphism. The kernel is exactly $\displaystyle \ker (\phi) = \text{SL}(n,q)$ thus we see by fundamental homomorphism theorem that $\displaystyle \text{SL}(n,q)$ is normal subgroup of $\displaystyle \text{GL}(n,q)$ and furthermore $\displaystyle \text{GL}(n,q)/\text{SL}(n,q)\simeq \mathbb{F}_q^{\times}$. Thus, $\displaystyle |\text{GL}(n,q)| = |\mathbb{F}_q^{\times}| |\text{SL}(n,q)|$ which means $\displaystyle (q^n-1)(q^n - q)...(q^n-q^{n-1}) = (q^n-1)|\text{SL}(n,q)|$. And so we have $\displaystyle \text{SL}(n,q) = (q^n-q)...(q^n-q^{n-1})$.
6)$\displaystyle \text{PGL}_n(F)$. Or as we seen $\displaystyle \text{PGL}(n,q)$. This $\displaystyle \text{GL}(n,q)/\text{Z}(\text{GL}(n,q))$ where $\displaystyle \text{Z}(\text{GL}(n,q))$ 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 $\displaystyle n$ and any $\displaystyle q$. 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)$\displaystyle D_n$. The symettries of a regular polygon of $\displaystyle n$ vertices. This group has size $\displaystyle 2n$. It is always a subgroup of $\displaystyle S_n$. It has size $\displaystyle 2n$. And it is always non-abelian when $\displaystyle n\geq 3$.