Named Groups

The Wolfram Language provides permutation representations for many important finite groups. Some of these groups are members of infinite families, parametrized by one or more integers; other groups are uniquely distinguished by their special properties and are frequently named after their discoverers.

This is the alternating group of degree 10:

In[1]:=1

✖

https://wolfram.com/xid/0jfziellsn-8bxqkz

Out[1]=1

Compute its order:

In[2]:=2

✖

https://wolfram.com/xid/0jfziellsn-i1k2vo

Out[2]=2

Give generators of an explicit permutation representation:

In[3]:=3

✖

https://wolfram.com/xid/0jfziellsn-67hwie

Out[3]=3

From these generators it is possible to reconstruct the group explicitly:

In[4]:=4

✖

https://wolfram.com/xid/0jfziellsn-5u3zfh

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/0jfziellsn-gqeann

Out[5]=5

The Wolfram Language provides information on the following infinite families of groups, and on some groups not belonging to parametrized families.

SymmetricGroup[n]	symmetric group of degree n
AlternatingGroup[n]	alternating group of degree n
CyclicGroup[n]	cyclic group of order n
DihedralGroup[n]	dihedral group of the n-gon, of order 2n
AbelianGroup[{n₁,n₂,…}]	Abelian group isomorphic to a direct product of several cyclic groups

Named infinite families of groups.

Mathieu Groups

The following five Mathieu groups were the first five sporadic simple groups to be discovered, in the second half of the nineteenth century, and are multiply transitive groups, all being subgroups of the largest one. The Wolfram Language provides default permutation representations for them.

MathieuGroupM11	first Mathieu group, acting on 11 points
MathieuGroupM12	second Mathieu group, acting on 12 points
MathieuGroupM22	third Mathieu group, acting on 22 points
MathieuGroupM23	fourth Mathieu group, acting on 23 points
MathieuGroupM24	fifth Mathieu group, acting on 24 points

Mathieu groups.

These are the orders of the five sporadic simple Mathieu groups:

In[6]:=6

✖

https://wolfram.com/xid/0jfziellsn-42ddl0

Out[6]=6

Explicit generators of a permutation representation of MathieuGroupM24 acting on 24 points:

In[7]:=7

✖

https://wolfram.com/xid/0jfziellsn-6b1hj9

In[8]:=8

✖

https://wolfram.com/xid/0jfziellsn-719qcl

Out[8]=8

To show that MathieuGroupM24 is 5-transitive, check transitivity of the group itself and of its first four stabilizers:

In[9]:=9

✖

https://wolfram.com/xid/0jfziellsn-jnibxv

Out[9]=9

In[10]:=10

✖

https://wolfram.com/xid/0jfziellsn-2a7xg7

Out[10]=10

In[11]:=11

✖

https://wolfram.com/xid/0jfziellsn-5l6rij

Out[11]=11

In[12]:=12

✖

https://wolfram.com/xid/0jfziellsn-gvzhru

Out[12]=12

In[13]:=13

✖

https://wolfram.com/xid/0jfziellsn-oezbfl

Out[13]=13

However, the stabilizer of five points is not transitive, because its action splits into two nontrivial orbits:

In[14]:=14

✖

https://wolfram.com/xid/0jfziellsn-m9rzkv

Out[14]=14

These are the orders of the stabilizers of the points of a base in the group. They correspond to the groups MathieuGroupM24, MathieuGroupM23, MathieuGroupM22, and then three more groups sometimes called Mathieu group

, Mathieu group

, and Mathieu group

, which are not simple. Finally there is the cyclic group of order 3 and the trivial group:

In[15]:=15

✖

https://wolfram.com/xid/0jfziellsn-ntszft

Out[15]=15

A similar chain is obtained starting from MathieuGroupM12. It contains MathieuGroupM11 and non-simple groups that can be called Mathieu group

, Mathieu group

, and Mathieu group

, with the trivial group at the end:

In[16]:=16

✖

https://wolfram.com/xid/0jfziellsn-tftwza

Out[16]=16

It is known that the largest permutation order in MathieuGroupM24 is 23. This is an example:

In[17]:=17

✖

https://wolfram.com/xid/0jfziellsn-18dxge

Out[17]=17

In[18]:=18

✖

https://wolfram.com/xid/0jfziellsn-jtidhz

Out[18]=18

In[19]:=19

✖

https://wolfram.com/xid/0jfziellsn-hkrod

Out[19]=19

Other Sporadic Simple Groups

There are 26 sporadic simple groups (27 if the Tits group is included). Apart from the five Mathieu groups, the Wolfram Language provides permutation representations for those of intermediate support length. The largest ones are too big to be handled as permutation groups in practice, and it is more efficient to represent them as matrix groups. These are the 13 groups (including the Tits group) for which representations on domains of less than 50000 points are known.

HigmanSimsGroupHS	Higman–Sims sporadic simple group
McLaughlinGroupMcL	McLaughlin sporadic simple group
JankoGroupJ1	Janko sporadic simple group
JankoGroupJ2	Janko sporadic simple group
JankoGroupJ3	Janko sporadic simple group
ConwayGroupCo2	Conway sporadic simple group
ConwayGroupCo3	Conway sporadic simple group
SuzukiGroupSuz	Suzuki sporadic simple group
HeldGroupHe	Held sporadic simple group
RudvalisGroupRu	Rudvalis sporadic simple group
FischerGroupFi22	Fischer sporadic simple group
FischerGroupFi23	Fischer sporadic simple group
TitsGroupT	Tits simple group

Intermediate sporadic simple groups.

Some sporadic groups are related to symmetries of the Leech lattice, a particular lattice in a Euclidean 24-dimensional space. These are sometimes known as the "second generation" of the sporadic simple groups.

The six groups of the second generation:

In[20]:=20

✖

https://wolfram.com/xid/0jfziellsn-w6ynm0

Their group orders are as follows:

In[21]:=21

✖

https://wolfram.com/xid/0jfziellsn-cfbhij

Out[21]=21

And these are the permutation degrees of the provided representations:

In[22]:=22

✖

https://wolfram.com/xid/0jfziellsn-s2ue6y

Out[22]=22

For example, these are generators for JankoGroupJ2 acting on 100 points:

In[23]:=23

✖

https://wolfram.com/xid/0jfziellsn-y1d76x

Out[23]=23

This is the last permutation in the group:

In[24]:=24

✖

https://wolfram.com/xid/0jfziellsn-cp8pi1

Out[24]=24

And this is its corresponding list of images:

In[25]:=25

✖

https://wolfram.com/xid/0jfziellsn-33cnkk

Out[25]=25

This is a chain of stabilizers of ConwayGroupCo2, acting on 2300 points. The base has six points only, and hence knowing the images of these six points suffices to uniquely identify each permutation in the group:

In[26]:=26

✖

https://wolfram.com/xid/0jfziellsn-9gicid

Out[26]=26

These are other sporadic simple groups representable as permutation groups on less than 50000 points:

In[27]:=27

✖

https://wolfram.com/xid/0jfziellsn-cc5b25

Their orders and degrees are as follows:

In[28]:=28

✖

https://wolfram.com/xid/0jfziellsn-30sh4

Out[28]=28

In[29]:=29

✖

https://wolfram.com/xid/0jfziellsn-smf07s

Out[29]=29

ONanGroupON	O'Nan sporadic simple group
HaradaNortonGroupHN	Harada–Norton sporadic simple group
ConwayGroupCo1	Conway sporadic simple group
FischerGroupFi24Prime	Fischer sporadic simple group
ThompsonGroupTh	Thompson sporadic simple group
JankoGroupJ4	Janko sporadic simple group
LyonsGroupLy	Lyons sporadic simple group
BabyMonsterGroupB	Baby monster sporadic simple group
MonsterGroupM	Monster group