Finite group presentation​
​by Oliver Seipel
Resources:
​https://en.wikipedia.org/wiki/Presentation_of_a _group​
​https://en.wikipedia.org/wiki/Finitely_generated_group​
​https://groupprops.subwiki.org/wiki/Groups_of_order _ 16​
​https://en.wikipedia.org/wiki/Semidirect_product​
​https://en.wikipedia.org/wiki/Direct_product​
​https://people.maths.bris.ac.uk/~matyd/GroupNames/​
​https://mathematica.stackexchange.com/questions/167787/how-do-i-write-semidirect-product-symbol-% e2 %8 b %8 a
The aim is to show group tables with generators:
Out[]=
1
a
2
a
3
a
b
ab
2
a
b
3
a
b
1
1
a
2
a
3
a
b
ab
2
a
b
3
a
b
a
a
2
a
3
a
1
ab
2
a
b
3
a
b
b
2
a
2
a
3
a
1
a
2
a
b
3
a
b
b
ab
3
a
3
a
1
a
2
a
3
a
b
b
ab
2
a
b
b
b
3
a
b
2
a
b
ab
2
a
a
1
3
a
ab
ab
b
3
a
b
2
a
b
3
a
2
a
a
1
2
a
b
2
a
b
ab
b
3
a
b
1
3
a
2
a
a
3
a
b
3
a
b
2
a
b
ab
b
a
1
3
a
2
a
A presentation of a group normally lookes like this:​
a,b
4
a
=1,
2
b
=
2
a
,ba=
3
a
b
​Here I use an Association with GroupOperationSymbol ∘ , type sc.​

4
a
1,
2
b

2
a
,b∘a
3
a
∘b
Remark: If
n
a
=1
then
-1
a
=
n-1
a
.
Definition: A presentation of a group is natural, if the product of the exponentials of the generators are the order of the group.
Examples:
​
Q8=
4
a
1,
2
b

2
a
,b∘a
3
a
∘b
​
The product of the exponentials is 4·2=8 which is the order of this group.
​
​
A4=<|
3
a
1,
2
b
1,
2
c
1,b∘aa∘c,c∘aa∘b∘c,c∘bb∘c|>
​
The product of the exponentials is 3·2·2=12 which is the order of this group.
​
​
S4=<|
4
a
1,
3
b
1,
2
c
1,b∘a
2
a
∘c,c∘aa∘b∘c,c∘b
2
b
∘c|>
​
The product of the exponentials is 4·3·2=24 which is the order of this group.
​
The presentations of A4 and S4 where found by the author 2017:)
Semidirect Product symbols ⋉, ⋊ (These are characters not Operators):
In[]:=
FullForm[⋉],FullForm[⋊]
Out[]=
⋉,⋊
This are the unicode codes and can be typed in directly:)
In[]:=
SetOptions[SelectedNotebook[],InputAliases->"spl"->"⋉","spr"->"⋊"]
Then you can use spl or spr to type them.
In[]:=
$verbose=True;(*turnonoroffverbosity*)

Mathematica code


Cyclic Group

In[]:=
cyclicGroup[{a_Symbol,n_}]:=<|
n
a
1|>
In[]:=
cyclicGroup[{a,n}]
Out[]=

n
a
1
In[]:=
C6=cyclicGroup[{a,6}]
Out[]=

6
a
1
In[]:=
CompactGroupTable[GroupPresentation[C6]]
Out[]=
1
a
2
a
3
a
4
a
5
a
a
2
a
3
a
4
a
5
a
1
2
a
3
a
4
a
5
a
1
a
3
a
4
a
5
a
1
a
2
a
4
a
5
a
1
a
2
a
3
a
5
a
1
a
2
a
3
a
4
a
The group table can be headed:
In[]:=
GroupTable[CompactGroupTable[GroupPresentation[C6]]]
Out[]=
1
a
2
a
3
a
4
a
5
a
1
1
a
2
a
3
a
4
a
5
a
a
a
2
a
3
a
4
a
5
a
1
2
a
2
a
3
a
4
a
5
a
1
a
3
a
3
a
4
a
5
a
1
a
2
a
4
a
4
a
5
a
1
a
2
a
3
a
5
a
5
a
1
a
2
a
3
a
4
a
The group table can be permuted:
In[]:=
CompactGroupTable[GroupPresentation[C6],"Permutation"->{1,4,2,3,5,6}]
Out[]=
1
3
a
a
2
a
4
a
5
a
3
a
1
4
a
5
a
a
2
a
a
4
a
2
a
3
a
5
a
1
2
a
5
a
3
a
4
a
1
a
4
a
a
5
a
1
2
a
3
a
5
a
2
a
1
a
3
a
4
a
The order of the elements can be shown:
In the first table the order of the elements are counted.
In the second table the order of each element is shown.
In[]:=
CompactGroupTable[GroupPresentation[C6],"Order"->True]
Order of the elements:
o
1
2
3
6
#
1
1
2
2
1
a
2
a
3
a
4
a
5
a
1
6
3
2
3
6
Out[]=
1
a
2
a
3
a
4
a
5
a
a
2
a
3
a
4
a
5
a
1
2
a
3
a
4
a
5
a
1
a
3
a
4
a
5
a
1
a
2
a
4
a
5
a
1
a
2
a
3
a
5
a
1
a
2
a
3
a
4
a
Definition: The center of a group is the set of all elements that commute with every element in a group, denoted
Z(G)={z:z·g=g·z,∀g∈G}
.
The center of a group can be shown:
In[]:=
CompactGroupTable[GroupPresentation[C6],"Center"->True]
center: {1,a,
2
a
,
3
a
,
4
a
,
5
a
}
commutative (abelian)
Out[]=
1
a
2
a
3
a
4
a
5
a
a
2
a
3
a
4
a
5
a
1
2
a
3
a
4
a
5
a
1
a
3
a
4
a
5
a
1
a
2
a
4
a
5
a
1
a
2
a
3
a
5
a
1
a
2
a
3
a
4
a
Lemma: If the center of a group has the order of the group, then and only then the group is abelian.

Direct Product of two Cyclic Groups

In[]:=
directProductCxC[{a_Symbol,n_},{b_Symbol,m_}]:=<|
n
a
1,
m
b
1,b∘aa∘b|>
In[]:=
directProductCxC[{a,n},{b,m}]
Out[]=

n
a
1,
m
b
1,b∘aa∘b
Remark: The group
C
3
×
C
2
is isomorphic to
C
6
.
In[]:=
C3xC2=directProductCxC[{a,3},{b,2}]
Out[]=

3
a
1,
2
b
1,b∘aa∘b
In[]:=
GroupOperationSymbol="";CompactGroupTable[GroupPresentation[C3xC2]]
Out[]=
1
a
2
a
b
ab
2
a
b
a
2
a
1
ab
2
a
b
b
2
a
1
a
2
a
b
b
ab
b
ab
2
a
b
1
a
2
a
ab
2
a
b
b
a
2
a
1
2
a
b
b
ab
2
a
1
a
In[]:=
GroupTable[CompactGroupTable[GroupPresentation[C3xC2],"Center"->True,"Order"->True,"Permutation"->{1,5,3,4,2,6}]]
center: {1,a,
2
a
,b,ab,
2
a
b}
commutative (abelian)
Order of the elements:

Semi Direct Product of two Cyclic Groups

An indexed group table:

Dihedral Group

Direct Product of a Dihedral and a Cyclic Group

Dicyclic Group

This is Hamilton’s Quaternion Group.

Direct Product of a Dycyclic and a Cylic Group

Direct Product of Cyclic Groups

Groups of order 4

This is the smallest non cyclic group:

Groups of order 6

This is the smallest non commutative group:

Groups of order 8

Groups of order 12

Groups of order 16

Source is missing :(
This should be an isomorphic group presentation:

SD16 (Semidihedral Group of degree 2)

M16 (Modular or Isanowa group of order 16)

SU2 (Group of the Pauli Matrices)

https://groupprops.subwiki.org/wiki/Central_product_of_D8_and_Z4

Small Group (16,3)

https://groupprops.subwiki.org/wiki/SmallGroup(16,3)

Is this a group of order 16? [Yes]

Frobenius 20

Frobenius 21

S4

All non-trivial elements have order 3.
https://en.wikipedia.org/wiki/List_of_small_groups
This could be isomorphic: