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.