Groups
A group is a nonempty set G equipped with a binary 
operation * that satisfies the following axioms:

1.	Closure:If a is in G and b is in G then a * b is in G.

2.	Associativity: a*(b*c) = (a*b)*c for all a, b, c in G.

3.	There is an element e in G called the identity element 
such that a*e = a = e*a for every a in G.

4.	For each a in G, there is an element d in G called the 
inverse of the element a such that a*d = d*a = e.

A group is said to be abelian if it also satisfies this 
axiom:

5.	Commutativity:  a*b = b*a for all a, b in G.

The order of a finite group is the number of elements in 
the group.  The order of an element is the least power of 
the element which = the identity.

S_3 is the symmetric group on 3 symbols.  It has order 6.
The elements of S_3 are 
{(1),(1,2),(1,3)(2,3),(1,2,3),(1,3,2)}.

The subset of S_3, { (1), (1,2,3), (1,3,2) ) also is a 
group.

D_4 is the group of translations of a square onto itself.  

D_4 consists of 4 rotations and 4 flips.

D_4 = {r_0,r_1,r_2,r_3,f_0,f_1,f_2,f_3}

Where
    r_0 = rotation of zero degrees counterclockwise.
    r_1 = rotation of  90  degrees counterclockwise.
    r_2 = rotation of 180  degrees counterclockwise.
    r_3 = rotation of 270  degrees counterclockwise.
    f_0 = flip around x-axis.( Called d in the textbook)
    f_1 = flip around y=x axis.(Called h in the textbook)
    f_2 = flip around y-axis.(Called t in the textbook)
    f_3 = flip around y=-x axis.(called v in the textbook)


The set {1, i, -1, -i } is a group under the operation of 
complex number multiplication.

The positive rational numbers form a group under real 
number multiplication.

The last two examples are special cases of:

In a ring with unity, the units of the ring form a group 
under the ring multiplication.

Z_n is the ring of integers mod n.  Z_n also refers to the 
additive group of this ring.

U_n is the group of units of Z_n.

GL(m,R) is the group of units in the ring of m by m 
matrices over the ring R.

An m by m matrix is a unit if and only if its determinant 
is nonzero.

Z_n has n elements.  The order of Z_n is n.

D_4 has 8 elements.  The order of D_4 is 8.

S_3 has 6 elements.  The order of S_3 is 6.

S_4 has 24 elements.  The order of S_4 is 24.

S_5 has 120 elements.  The order of S_5 is 120.

S_n has n! elements.  The order of S_n is n!.

If G is a group, and b is an element in G, the the set 

H = { b^k such that k is an integer} is a subgroup of G.

Indeed, if G is only a semi-group then

H = { b^k such that k is an integer} is a subset of G which 
is a group under the semi-group operation.

Note:  a semi-group requires only the closure and 
associativity axioms.


U_3 has 2 elements.  The order of U_3 is 2.

U_4 has 2 elements.  The order of U_4 is 2.

U_5 has 4 elements.  The order of U_5 is 4.

U_6 has 2 elements.  The order of U_6 is 2.

U_7 has 6 elements.  The order of U_7 is 6.

U_8 has 4 elements.  The order of U_8 is 4.

U_9 has 6 elements.  The order of U_9 is 6.

U_10 has 4 elements.  The order of U_10 is 4.

U_11 has 10 elements.  The order of U_11 is 10.

U_12 has 4 elements.  The order of U_12 is 4.

U_n has as number of elements the number of positive 
integers < n that are relative prime to n.

Define phi(n) = number of positive integers less than n 
that are relative prime to n.

Order(U_n) = phi(n).

Let f be a 1-1 map of the elements of a group onto itself.  
Define a new group operation @ by

A @ B =  f^-1 (  f(A) * f(B) ).  

Then G is also a group under the new operation @.

The bi-jective functions of a set onto itself form a group 
under the operation of composition of functions.

SL(m,R) is the group of m by m matrices over the ring R 
such that their determinant = 1.

In the operation table for a finite group, each element 
appears exactly once in each row and column.

The set of linear functions over the reals form a group 
under the operation composition of functions.