= Group
{wiki=Group_(mathematics)}
= Group
{disambiguate=mathematics}
{synonym}
= Center
{disambiguate=group theory}
{parent=Group}
{wiki}
= Center
{disambiguate=group}
{synonym}
= Group axiom
{parent=Group}
= Commutative property
{parent=Group axiom}
{wiki}
= Commutative
{synonym}
= Commutativity
{synonym}
= Abelian group
{c}
{parent=Commutative property}
{wiki}
= Abelian
{c}
{synonym}
Easily classified as the [direct product] of of [prime] order.
= Non-commutative
{parent=Abelian group}
= Symmetry
{parent=Group}
{wiki}
Directly modelled by .
For , see: .
= Symmetry breaking
{parent=Symmetry}
{wiki}
= Important mathematical group
{parent=Group}
= Important discrete mathematical group
{parent=Important mathematical group}
= Cyclic group
{parent=Important discrete mathematical group}
{wiki}
= $C_n$
{synonym}
{title2}
= The direct product of two cyclic groups of coprime order is another cyclic group
{parent=Important discrete mathematical group}
{wiki}
You just map the value (1, 1) $C_m \times C_n$ to the value 1 of $C_{mn}$, and it works out. E.g. for $C_2 \times C_3$, the [group generated by] of (1, 1) is:
``
0 = (0, 0)
1 = (1, 1)
2 = (0, 2)
3 = (1, 0)
4 = (0, 1)
5 = (1, 2)
6 = (0, 0) = 0
``
= Permutation
{parent=Important discrete mathematical group}
{wiki}
= Cycle notation
{parent=Permutation}
{wiki}
A concise to describe a specific .
A permutation group can then be described in terms of the of specific elements given in cycle notation.
E.g. https://en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Permutation_groups mentions that the is generated by three elements:
* (0123456789a)
* (0b)(1a)(25)(37)(48)(69)
* (26a7)(3945)
which feels quite compact for a with 95040 elements, doesn't it!
= Parity of a permutation
{parent=Permutation}
{wiki}
= Odd permutation
{parent=Parity of a permutation}
= Even permutation
{parent=Parity of a permutation}
= Permutation group
{parent=Permutation}
{wiki}
= Stabilizer
{disambiguate=group}
{parent=Permutation group}
Suppose we have a given that acts on a set of n elements.
If we pick k elements of the set, the stabilizer subgroup of those k elements is a subgroup of the given permutation group that keeps those elements unchanged.
Note that an analogous definition can be given for non-finite groups. Also note that the case for all finite groups is covered by the permutation definition since
TODO existence and uniqueness. Existence is obvious for the identity permutation, but proper subgroup likely does not exist in general.
Bibliography:
* https://mathworld.wolfram.com/Stabilizer.html
* https://ncatlab.org/nlab/show/stabilizer+group from
= Symmetric group
{parent=Permutation group}
{wiki}
of all .
= All groups are isomorphic to a subgroup of the symmetric group
{parent=Symmetric group}
Or in other words: are boring, because they are basically everything already!
= Alternating group
{parent=Permutation group}
{wiki}
= $A_n$
{synonym}
{title2}
Group of .
Note that don't form a of the like the even permutations do, because the composition of two odd permutations is an even permutation.
= Alternating group of degree 5
{parent=Alternating group}
= The alternating groups of degree 5 or greater are simple
{parent=Alternating group of degree 5}
https://www.youtube.com/watch?v=U_618kB6P1Q GT18.2. A_n is Simple (n ge 5) by (2012)
= Dihedral group
{parent=Important discrete mathematical group}
{title2=$D_n$}
{wiki}
Our notation: $D_n$, called "dihedral group of degree n", means the dihedral group of the with $n$ sides, and therefore has order $2n$ (all rotations + flips), called the "dihedral group of 2n".
= Wallpaper group
{parent=Important discrete mathematical group}
{wiki}
17 of them.
= Space group
{parent=Important discrete mathematical group}
{tag=Crystallography}
{wiki}
All possible repetitive crystal structures!
219 of them.
= Klein four-group
{c}
{parent=Important discrete mathematical group}
{wiki}
$C_2 \times C_2$
= Finite group
{parent=Group}
= Classification of finite groups
{parent=Finite group}
As shown in