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Groups
What are groups? A group is simply a set of members and one operation • which is performed on two members of the group, generating a third. There are some other conditions:
Associativity: For all a, b, c in the group, (a•b)•c = a•(b•c)
Identity: There is an element i in the group such that i•a = a and a•i = a.
Inverse: For all a there is an element a-1 such that a•a-1 = i and a-1•a = i.
This implies another result: that for all a, b there is an element c such that a•c = b. Proof: b = i•b. Let there be a d such that d = a-1. Then c is d•b. Thus a•c = a•(d•b) = (a•d)•b = (a•a-1)•b = i•b = b.
A subgroup is a set of members of a group that form a group themselves. The non-trivial subgroups are all subgroups except i and the entire group. A subgroup H of group G is normal if for all a in G and b in H, (a•b)•a-1 is in H. Some groups are simple. These are like prime numbers, as all groups can be made out of simple subgroups. Simple groups have no non-trivial normal subgroups.
Abelian groups
There is a type of group called abelian groups (named after Niels Hendrik Abel). If a group is abelian then for all a, b a•b = b•a. Abelian groups include all cyclic groups. These are groups formed by modular addition. Let ab for an element a and a natural number b be defined as following:
a1 = a
ab+1 = a•ab
The group of all elements of the form ab is the subgroup generated by a. This group is cyclic and therefore abelian. One theorem is that if a subgroup generated by an element a is the entire group then the group is abelian. The converse is not necessarily true, for example in my Colour Octonion group.
Symmetry groups
Some groups are symmetry groups. The group of any polygon with no lines of symmetry is cyclic, as the only symmetries are rotational. However, the symmetry group of a regular polygon is more complex, as reflections must also be accounted for. With polyhedrons it is more complex still. However, the cube and octahedron share the same symmetry group, as do the dodecahedron and icosahedron. It would be even more complex in four dimensions. If infinite groups are allowed, translational symmetry becomes possible.