group
CAIN
Formally, a group G is defined to be a set with an associative binary operation ⋆, called product.
We write [G, ⋆] as the group if G satisfies the group properties in CAIN
(Closed, Associative, Identity, iNverses)
abelian
A group G is Abelian or Commutative if a ⋆ b = b ⋆ a for each a, b ∈ G. (A B E L)
In general in a group a ⋆ b != b ⋆ a.
C in CAIN does not stand for commutative.
binary operation
A binary operation ⋆ on set S is a mapping ( function) that assigns to each ordered pair or (a, b) ∈ S × S some unique element of S.
We write a ⋆ b for ⋆(a, b) or just ab.
closed
group property
If the domain of ⋆ is S ×S and the co-domain of ⋆ is S, we say that S is closed under the operation ⋆.
⋆(a,b) = a⋆b ∈S for all a,b∈S
associative
group property
The operation is associative if (a ⋆ b) ⋆ c = a ⋆ (b ⋆ c) for a, b, c ∈ S.
All groups have an associative binary operation.
⋆ is an associative binary operation
identity
group property
An element e in S is called the Unity, Neutral element or Identity for the operation ⋆ if a ⋆ e = e ⋆ a = a for all a in S.
All groups have a unique identity.
inverse
group property
An element a in S is said to be a Unit in S or is Invertible in S under ⋆ if we can find an inverse for a in S. That is, there is b ∈ S such that a ⋆ b = b ⋆ a = e, where e is unity in S. In a group all elements are invertible and hence are units.
b is unique and can be written a^(-1).
inverses
proposition (5)
Let G be a group. Then
1. If e is unity then e−1 = e.
2. If a ∈ G then the inverse of a is unique. We write this unique element as a−1.
Further we have that (a−1)−1 = a.
3. If a and b are in G then so is a ⋆ b and (a ⋆ b)−1 = b−1 ⋆ a−1.
4. If a1, a2, · · · , an are units so is a1 ⋆ a2 ⋆ · · · ⋆ an and
(a1 ⋆ a2 ⋆ · · · ⋆ an)−1 = a−1
n ⋆ · · · ⋆ a2−1 ⋆ a1−1.
5. If a is a unit so is an for any n ∈ Z where a0 = e, and (a−1)n = (a−1)n.
cancellation laws
proposition
Let g, h and f be in group G, then
(i) If gh = gf, then h = f.
(ii) If hg = fg, then h = f.
can’t cancel gh=fg unless G is abelian
unique solutions
proposition
Let h and g be in G then
(i) The equation gx = h has a unique solution x = g−1h in G.
(ii) The equation xg = h has a unique solution x = hg−1.
subgroup
definition
A non-empty subset H of G is a subgroup of G, if H itself forms a group under the same operation as G. We may write H ≤ G.
Each group has two subgroups.
1. H = {e} - the trivial subgroup
2. H=G - the* improper subgroup.*
Other subgroups are called
proper subgroups of G.
Here we write H < G.
subgroup
theorem
Let H be a non empty subset of G. Then H is a subgroup of G if and only if ab^(−1) ∈ H for all a, b ∈ H.
first show H is nonempty
order/cardinality
groups
|G| is the number of elements in G.
The order of an element g in G is the smallest positive integer n such that gn = e.
May be finite of infinite.
(Difference between order of group G and element g∈G)
cayley tables
commutative - symmetry about main diagonal (abelian structure)
associative - symmetry of other diagonal
closed - each element appears only once in each row/column (unity e links inverses)
