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)

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-1b

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