__Group Axioms__
Though we describe the definition of a group by axioms,
simple though they are, some groups can be almost impossible to imagine
whole. Mathematicians consider the internal structure of subgroups
inside groups, and study their type and composition.
The axioms are as follows.
A group is a set ‘G’, coupled with a binary* operation ‘+’, such that
the following conditions are satisfied. (*binary, in the sense that the
operation '+' is a map from member pairs of G to a single member of ‘G’.
Therefore ( g , h ) => k. More commonly we would consider this as, g +
h = k, or gh = k)
1) Axiom of Closure
For all elements g and h in G, (g + h) is also in G.
2) Axiom of Associativity
For all elements a, b and c of G under the binary operation ‘+’ of G,
a +(b + c) = (a + b) + c
3) Axiom – Existence of an identity element in G.
There must exist a unique identity element ‘e’, such that for all
elements g in G,
e + g = g + e = g
4) Axiom – Existence of inverses
For every element g in G there must exist a unique element h in G so
that,
g + h = h + g = e
where e is the identity element. ‘h’, without loss of generality may be
written ‘-g’ (minus g)
__Note__
There is no reason that ‘+’ must be commutative in G, so that g + h = h
+ g. There are many groups where this is not the case. It is customary
when the operation is not commutative, to use multiplication instead of
addition. Mathematicians are uncomfortable with a + b not being b + a.
It is still tradition to retain ‘e’ as an identity element.
Mathematicians, instead of writing inverses in multiplicative notation
as 1/g, or even e/g, they write g^{-1} a
A group G with a commutative operation ‘+’ is called “Abelian”. It is a
requirement that a + b = b + a, for ANY two elements of such a group.
Obviously, groups without a commutative operation are termed
“Non-Abelian”.
Common examples
are...
The set of
integers under normal addition is a group. It is an infinite abelian
group.
The set of
symmetries of a regular polygon is a group. There are n rotational
symmetries for an n-gon, as well as the n rotations of a reflection.
(the reflection on its own is it's own inverse). The identity element,
'e' is a rotation of one full turn, or indeed, no turn at all.
Therefore, for an n-gon, the group D(2n) has 2n elements. This group is
finite and non-abelian.
__Useful Fact.
__
Because the existence of inverses is unique, the cancellation laws apply
for elements with inverses. This means that ab = ac implies directly
that b = c, (multiply on left by the inverse of a). In a group, this is
equivalent that a solution to the equation ax = b exists within the
group. In fact it is simply x = a^{-1}b
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