Group Theory Notes

By Xah Lee. Date: . Last updated: .

Group Definition

a group, written as (G,*), is a set G and a binary operation * such that the following 3 axioms hold:

  1. for all x y in G, x * y is in G.
  2. for any x y z in G, (x * y) * z == x * (y * z)
  3. There exist a element e, such that e*x == x, and x*e == x.

As a consequence of group axioms:

trivial group

a trivial group is a group consisting of a single element. That element is the identity element.

order of a group

the order of a group is the number of elements in the set.

group homomorphism

a function f from a group G to another group H such that

f(x*y) == f(x) * f(y)


A set H is a subgroup of a group G if it is a subset of G and is a group using the operation defined on G.

note: the identity element, and invers elements, of G, are in H.

Note that 2 subgroups may overlap. That is, share elements and different in size and one is not a subset of the other. e.g. group of addition modulo 8, with member 0 to 7. {0,4} is a subgroup, so is {0,2,4,6}


if G is a group, and H is a subgroup of G, and g is a element of G, then

So, a coset is defined with respect to: (1) a subgroup of G, (2) a element in G.

any 2 left cosets are either identical or don't have common element. So, all left cosets is a partition of G.

Same for righ cosets.

normal subgroup

Let G be a group, and H be a subgroup of G.

the subgroup H is a “normal subgroup” when:

if the set of all left cosets gH, and the set of all right cosets Hg, are the same, for all g in G.

quotient group

Let N be a normal subgroup of a group G. We define the set G/N to be the set of all left cosets of N in G, i.e., G/N = { aN : a ∈ G }. Define an operation on G/N as follows. For each aN and bN in G/N, the product of aN and bN is (aN)(bN). This defines an operation on G/N if we impose (aN)(bN) = (ab)N, because (ab)N does not depend on the choice of the representatives a and b: if xN = aN and yN = bN for some x, y in G, then

(ab)N = a(bN) = a(yN) = a(Ny) = (aN)y = (xN)y = x(Ny) = x(yN) = (xy)N.

Here we have used in an important way that N is a normal subgroup. One checks that this operation on G/N is associative, has identity element N, and the inverse of a element aN of G/N is a−1N. Therefore, the set G/N together with the operation defined above forms a group; this is known as the quotient group of G by N.

Because of the normality of N, the left cosets and right cosets of N in G are equal, and so we could have instead defined G/N to be the set of right cosets of N in G.

simple group

a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated.

permutation group, symmetry group

a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself).

The group of all permutations of a set M is the symmetric group of M, often written as Sym(M).[1] The term permutation group thus means a subgroup of the symmetric group. If M = {1,2,...,n} then, Sym(M), the symmetric group on n letters is usually denoted by Sn.

The way in which the elements of a permutation group permute the elements of the set is called its group action.

the degree of a permutation group is the number of elements the permutation is on.


one most simple and clear notation is so called cayley two-line notation. like this

1 2 3 4 5
3 5 2 4 1

This means, this action, is to map 1st elment to 3rd, 2nd to 5th, 3rd to 2nd, 4th to 4th (no change), 5th to 1st.

It can also be written in “cyclic notation”. For example, the above permutation can be written as

(1 3 2 5) (4)

meaning, 1 mapped to 3, 3 mapped to 2, 2 mapped to 5. And 4 mapped to 4. (the last one, 4, can be dropped, since it's no change.)

direct product of a group

If (G,*) and (H,•) are groups, then the set G×H together with the operation (g1,h1)(g2,h2) = (g1*g2,h1•h2) is a group.

The diff between ismorphism and homomorphism is that the former is bijection.

Given two groups (G, *) and (H, @), a group isomorphism from (G, *) to (H, @) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function f : G → H such that for all u and v in G it holds that

f (u * v) = f (u) @ f (v)

The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order. Lagrange's theorem (group theory)


free group

rotation group SO(3)

galois theory

combinatorial group theory

geometric group theory


Let's say we have a set S of 3 elements {a,b,c}, and we have a binary operation * on it. We can then have a mulitplication table, such as:

   a b c
 a a b c
 b b a b
 c c b a

now, if we draw all possible multiplication table for binary operation on S. Then, put those multiplication table on two sides, one that satisfies the associative law, and one not.

What can we discern from the patterns on the multiplication table, about the associative property?

write a program, that generats all groups of nth order, and display their multiplication table in HTML as html tables, with each group element color coded.

for a group of 3 elements, the there are 9 entries in the table (9 = 3*3).

consider all possible multiplication tables, the numbers are: 3^9. 19683.

2^(2*2) = 16.

possible binary combination is n*n. Each entry has n possibilities, multiply that by all possible binary combination, we have n^(n*n)

n=1 , 2, ... elements are 1, 10, 3330, 178981952, ... (Sloane's A001329),


Representation of a magma

Each group elements will be represented by integers starting with 1. A list of 3 numbers will represent a binary operation. e.g. (5 2 3) will mean 5*2=3.

A group can be represented as a list of binary operations. For example, a group of 3 elements:

(1 1 n1)
(1 2 n2)
(1 3 n3)
(2 1 n4)
(2 2 n5)
(2 3 n6)

where a element, e.g. (1 1 n1) means 1*1:=n1.

Seeing how only the n1, n2... is unique for a group, we represent a group by a list of integers, each represent the product of 1*1, 1*2,..., 2*1,2*2,... etc. For example, a group of 3 elements:

((m11 m12 m13 ...) (m21 m22 m23 ...) ...)

each row of the mult-table, is captured by a sublist. Each column, is a sublist of the transpose list.

restriction to the group list

we require that the structure has a identity.

This means, when the group is represented as a table, one of the rows should be identical to the top horizontal group elements, and one of the column should be identical to the left side group elements.

For simplicity, we represent identity using “1”. This means, our group will start like this:

(1 2 3 ... 2 m22 m23 ... 3 ...)

if n is the length of the list, then the order of the group is Sqrt[n].

detection of magma properties

• Identity to determin if a magma contains a identity element, just check if any one of the row has the same element as the row title, and similarly there must be such a column.