### Algebraic definition

A set G equipped with operation $\bullet$

> Closure: $\forall a,b \in G, a \bullet b \in G$  
>Associativity: $(a \bullet b) \bullet c = a \bullet (b \bullet c)$  
>Identity element: $\exists e \in G, such \ that \ \forall a \in G \ a \bullet e = a$  
>Inverse element: $\forall a \in G, \exists a^{-1}, such \ that \ a \bullet a^{-1} = e$

Can be shown, that $e \bullet a = a$ and $a^{-1} \bullet a = e$ and inverse and e are unique

**Abelian group**: $\forall a,b \in G, a \bullet b = b \bullet a$

If a group is abelian, usually group operation is denoted by + and the identity element by 0

As usual in algebra are difined Homomorphism, Isomorphism, Endomorphism, Automorphism. In Algebra isomorph groups are deemed the same.

group homomorphism $f:G \to H$ such that $f(g_1 \bullet g_2) = f(g_1) * f(g_2)$

**kernel** of a homomorphism is defined as the set of elements that get mapped to the identity element in the image.

Group operation may be viewed as bijection from G to G.

**Subgroup**: if H subset of G, which is closed under group operation and inverse operation. $H \leq G$, proper subgroup: $H \lt G$

&nbsp;

Finite groups can be described by **Cayley table**:

$
\begin{bmatrix}
& Elmnts & e & a & b & c  \\
& e & e & a  & b & c  \\
& a & a & a \bullet a & a \bullet b & a \bullet c  \\
& b & b & b \bullet a & b \bullet b & b \bullet c  \\
& c & c & c \bullet a & c \bullet b & c \bullet c  \\
\end{bmatrix}
$

Group can be described by **Presentation of a group**:  

$\langle S \vert R  \rangle$ where S is set of generators and R is a set of rules

free group $F_S$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of S: $\langle S \vert \emptyset  \rangle$



### Normal Subgroup

**Coset left** (right): if $H \lt G$, set defined by an element $gH = \{g \bullet h : h \in H\}$ ($Hg = \{h \bullet g : h \in H\}$)

cosets form a partitioning of group, any element of a coset is a represantative of the coset. the number of distinct cosets is called index of subgroup. 

if $H \lt G$ then $ index= \frac {\vert G  \vert} {\vert H  \vert} $ . A finite groups  of prime size can not have subgroups.

if $\forall g \in G : gH = Hg$, then $H$ is said to be a normal subgroup. Subgroups of  abelian group are normal.

**Conjugacy class**: $Cl(a)=\{ b \in G : \exists g \in G \text{ such that } b = gag^-1\}$

Conjugacy class is an equivalence relation and thus partitions the group. For abelian group $Cl(a) = \{a\}$

group of **Inner automorphisms**; $Inn(G)$
>$\varphi_g: G \rightarrow  G$ such that $\varphi_g(x)=gxg^-1$  
> $Inn(AbelianGroup) \simeq \{e\}$  
>Normal subgoup is preserved by inner automorphisms.  
>kernel of a homomorphism is equal to a normal subgroup

**Quotient group (factor group)**: we can define a group operation on cosets as the following $g_1H*g_2H=(g_1 \bullet g_2)H$. 

If $H \vartriangleleft G$ then  $G/H$ is a group. ( $G/\{e\}=G$ , $G/G=\{e\}$ )

&nbsp;

**Direct product**: $G \times H$ is the ordered pairs $(g,h)$ where $g \in G$ and $h \in H$ such that $(g_1,h_1) \bullet (g_2,h_2) == (g_1 * g_2, h_1*'h_2)$

Let a group $P$ has 2 subgroups $G,H$. 

$
P=G \times H 
\Leftrightarrow 
\begin{cases}
 & G \cap H == \{e\} \\ 
 & \text{ every element can be expressed in } g \bullet h \\ 
 & G \text{ comutes with } H \text{( or stronger:} G \text{ and } H \text{ are normal subgroups} \\
\end{cases}
$




### Geometric definition

**Transformation group (Group action)** : $\varphi$ is a function of G on set X

$\varphi : G \times X \rightarrow X:(g,x) \mapsto \varphi(g,x) $ where $\varphi(e,x)=x$, and $\varphi$ is bijective map.

**symmetric group**: defined over any set is the group whose elements are all the bijections from the set to itself. For finite set of size n, it is $S_n$: all permutations of finite set of size n.

**Cayley's theorem**: every group can be realized as a subgroup of a symmetric group over some set

**Orbit**: Suppose $G$ is a group acting on a set $S$. ${G_{.s}}$ the orbit of $s \in S$ is defined as $\{t \in S \vert \exists g \in G, g.t=s \}$

The set of orbits of (points x in) X under the action of G form a partition of X.

A group action is termed **transitive** if it has exactly one orbit: $\forall x,y \in S, \exists g \in G , g.x = y$

**Stabilizer (isotropy group)**: denoted $Stab_G(s)$, is defined as: $\{ h \in G \mid \ h.s = s \} $.  

Actions of groups on vector spaces are called **representations** of the group 

Alternativly a **representation** of a group G on a vector space V is a group homomorphism $\varphi: G \rightarrow GL(V,F)$

**General linear group** : $GL(V,F)$ are all matrices over a field $F$, with 0 determinant (invertible).
> $SL(n, F)$: determinant is 1. In $R^n$  volume and orientation preserving linear transformations  
> $O(n)$: orthogonal matrcies $O^TO=I$. Preserves the dot product of vectors.  
>$SU(n)$: n×n unitary matrices with determinant 1 on $\mathbb{C}$


A **faithful representation** is one in which the homomorphism G → GL(V) is injective; in other words, one whose kernel is the trivial subgroup {e} consisting only of the group's identity element.








### examples

small groups of order n:
> n=1, the only  group {e} == **trivial group**

> n=2, only one group $Z_2 \simeq S_2$  
>> $
\begin{bmatrix}
&    & e & a \\
& e & e & a \\
& a & a & e \\
\end{bmatrix}
$

> n=3, only one group $Z_3$
>>$\begin{bmatrix}
&    & e & a & b \\
& e & e & a & b \\
& a & a & b & e\\
& b & b & e & a\\
\end{bmatrix}$  
>> that is $b = a \bullet a == a^2$

> n=4, two non isomorphic groups
>> $Z_4$  
>>>$\begin{bmatrix}
&    & e & g & g^2 & g^3 \\
& e & e & g & g^2 & g^3 \\
& g & g & g^2 & g^3 & e\\
& g^2 & g^2 & g^3 & e & g\\
& g^3 & g^3 & e & g & g^2\\
\end{bmatrix}$  
>> this can be viewed as a group of multiplications of $\{1, i, -1, -i\}$  
>> one proper subgroup $Z_2 \lt Z_4 : \{ e, g^2 \}$  
>> this subgroup partitions the group to the following cosets: $\{e, g^2\}$, $\{g, g^3\}$

>> Klein four-group $K_4$: 
>>>$
\begin{bmatrix}
&  & e & a & b & c  \\
& e & e & a  & b & c  \\
& a & a & e & c & b  \\
& b & b & c & e & a  \\
& c & c & b & a & e  \\
\end{bmatrix}
$
>> 3 proper subgroups : $\{ e, a \},\{ e, b \},\{ e, c \}$  
>> $K_4 = Z_2 \times Z_2 $ and can be presented as $ \langle a, b \vert a^2,b^2, ab=ba  \rangle$


**symmetric group**:  $S_3$

> order of the group is $3!=6$  
> cycle examples:
>>$(1,2) == \begin{pmatrix}
1 & 2 & 3\\ 
2 & 1 & 3
\end{pmatrix}$  
>>$(1,2,3) == \begin{pmatrix}
1 & 2 & 3\\ 
2 & 3 & 1
\end{pmatrix}$  
> elements (in cycles):  { e, (1,2), (1,3), (2,3), (1,2,3), (1,3,2) }  
> first non abelian group:  (1,2)(2,3) = (1,2,3) and (2,3)(1,2)=(1,3,2)  
>presentations: $\langle a, b, c \vert a^2,b^2, c^3, abc  \rangle$ or $\langle s_1, s_2 \vert s_1^2,s_2^2, (s_1s_2)^3  \rangle$ or $\langle s, t \vert s^3,t^2, (st)=ts^2  \rangle$

> proper subgroups: 
>>$Z_2$: {e,(1,2)}, {e,(1,3)}, {e,(2,3)}  
>> $Z_3$: {e,(1,2,3), (1,3,2)}  
> conjugacy class partitioning
>> {e}  
>>{ (1,2), (1,3), (2,3) }  
>>{ (1,2,3), (1,3,2) } this is the only normal subgroup

>$Inn(S_3) \simeq S_3$

**Cyclic group** of order n:  $Z_n$ : $\langle a \vert a^n  \rangle$, that is one generator and $a^n = e$
> $Z_n$ is abelian group

> it may be looked as 
>> modulo n arithmetics  
>> rotations by $2\pi \frac {m} {n}$

**Free group** of order 1:  $F_1: \langle a \vert \emptyset  \rangle$
> consists of all possible strings of $a$ and $a^{-1}$ reducted by the group identity $a \bullet a^{-1} = e$  
> for example $"aaaaaa"$ or $"a^{-1}a^{-1}a^{-1}"$  
>  $F_1 \simeq Z$ where $Z$ with + is treated as Cyclic goup


## example  Dihedral transformation group 
$D_n$ is the group of symmetries of a regular polygon (includes rotations and reflections).

Orbit of a point in $R^2$ is the set of vertexes of a polygon.

$D_n = \langle r,s \vert r^n, s^2, (sr)^2  \rangle$, order of the group = 2n,  $D_3 \simeq S_3$

$(sr)^2=e \Leftrightarrow srs=r^{-1}$

$r_i==r^i; s_i==r_is \Rightarrow r_ir_j=r_{i+j}, r_is_j=s_{i+j},s_jr_i=s_{i-j},s_is_j=r_{i-j}$ 

natural represantation in $R^2$: 
$
r_i=\begin{pmatrix}
cos \frac{2 \pi i}{n} & -sin \frac{2 \pi i}{n}\\ 
sin \frac{2 \pi i}{n} & cos \frac{2 \pi i}{n}
\end{pmatrix}
,
s_i=\begin{pmatrix}
cos \frac{2 \pi i}{n} & sin \frac{2 \pi i}{n}\\ 
sin \frac{2 \pi i}{n} & -cos \frac{2 \pi i}{n}
\end{pmatrix}
$

$D_4 \lt O(2)$, $D_4 \lt SO(3)$

$D_4$: group of symmetries of a square

 

> conjucacy classes:
>> $\{e\}$  
>> $\{r^2\}$ rotation by $\pi$  
>> $\{s,s_2\}$ reflections horisontal and vertical  
>> $\{s_1,s_3\}$ diagonal reflections  
>> $\{r,r^3\}$ rotations by $\pi/2$ and $3\pi/2$

> proper subgroups:
>> $\{e,r^2\}$ Normal subgroup  
>> $\{e,s\}$,$\{e,s_1\}$,$\{e,s_2\}$,$\{e,s_3\}$  
>> $\{e,s,r^2,r^2s\}$, $\{e,rs,r^2,r^3s\}$  
>> $\{e,r,r^2,r^3\}$  Normal subgroup

> $Inn(D_4) \simeq K_4$


```python
# some sympy Permutation examples

# this is needed for google.colab
#from IPython.display import HTML
#display(HTML(""))

from sympy import init_printing
init_printing(use_latex='mathjax')

import sympy.combinatorics.permutations as P

p1 = P.Permutation(4)
display(p1.array_form)
display(p1)  #p1.cyclic_form
print('='*50)

p1 = P.Permutation(3,1)(0,2,4)
display(p1.array_form)
display(p1)
print('='*50)

p2=p1*p1
display(p2.array_form)
display(p2)
print('='*50)

```


$$\left [ 0, \quad 1, \quad 2, \quad 3, \quad 4\right ]$$



$$\left( 4\right)$$


    ==================================================



$$\left [ 2, \quad 3, \quad 4, \quad 1, \quad 0\right ]$$



$$\left( 0\; 2\; 4\right)\left( 1\; 3\right)$$


    ==================================================



$$\left [ 4, \quad 1, \quad 0, \quad 3, \quad 2\right ]$$



$$\left( 0\; 4\; 2\right)$$


    ==================================================



```python
p1('abcde')
```




    ['c', 'd', 'e', 'b', 'a']




```python
import sympy.combinatorics.generators as G
d = G.dihedral(4)
list(d)
```




$$\left [ \left( 3\right), \quad \left( 0\; 3\right)\left( 1\; 2\right), \quad \left( 0\; 1\; 2\; 3\right), \quad \left( 1\; 3\right), \quad \left( 0\; 2\right)\left( 1\; 3\right), \quad \left( 0\; 1\right)\left( 2\; 3\right), \quad \left( 0\; 3\; 2\; 1\right), \quad \left( 0\; 2\right)\left( 3\right)\right ]$$




```python
#S3
from sympy.combinatorics.free_groups import free_group, FreeGroup
import sympy.combinatorics.fp_groups
F, a, b = free_group("a, b")
G = FpGroup(F, [a**2, b**3, (a*b)**4])
print("Order is:",G.order())
#l = sympy.combinatorics.fp_groups.low_index_subgroups(G,2)
#for t in l:
#    print(t.table)
```

    Order is: 24


## Links

[The Group Properties Wiki](https://groupprops.subwiki.org/wiki/Main_Page)  

[Magma Computational Algebra System](http://magma.maths.usyd.edu.au/magma/)

[GAP - Groups, Algorithms, Programming -a System for Computational Discrete Algebra](https://www.gap-system.org/)

http://www.sagemath.org/

[Handbook of Computational Group Theory. Derek F. Holt, Bettina Eick, Eamonn A. O'Brien](https://books.google.am/books?id=rnTLBQAAQBAJ&source=gbs_book_other_versions)

https://docs.sympy.org/latest/modules/combinatorics/perm_groups.html

https://people.maths.bris.ac.uk/~matyd/GroupNames/index.html


```python

```

14163f558b14909503e6c3d2a3149d33b04abbdb
ipynb
Jupyter Notebook
Math/IntroToGroups1.ipynb
gate42qc/seminars
35ff77b902d9c2ede619fd6e2d9c3e80d20d78de
MIT
2018-12-07T10:02:06.000Z
2019-11-24T19:30:03.000Z
Math/IntroToGroups1.ipynb
gate42qc/seminars
35ff77b902d9c2ede619fd6e2d9c3e80d20d78de
MIT
Math/IntroToGroups1.ipynb
gate42qc/seminars
35ff77b902d9c2ede619fd6e2d9c3e80d20d78de
MIT
2019-08-22T12:07:40.000Z
2019-08-22T12:07:40.000Z
Qwen/Qwen-72B
1. YES
2. YES