Introduction
Maths, can help more than it confuses. If it does so, then it is done well. God, from the Trinitarian perspective and including Anselm's argument can never be "second best" in the greatness stakes, else he wouldn't be God. Each member of the trinity must be fully as great as the others, maybe in different ways, but still there must be a symmetry involved. When one member is compared with respect to the others there must be equality, though the quality may indeed take different forms.
Abstract Algebra - Hang on, those x's and y's? Well, yes and no. The use of algebra to describe symmetries, like reflection in a mirror, or the number of times in a whole revolution that a shape will fit upon itself as if never moved, can be modelled with algebra. The algebra is called "abstract" because there is in general no numerical "answer" other than a representation of a structure's symmetries. There are many different types of structure that have symmetry, many of which are too complicated to imagine. Algebra may describe objects in any number of dimensions, and is often the only way we can attempt to understand such multidimensional structures, through their symmetries.
The Algebraic structures I am interested in are 'Groups', 'Rings' and 'Fields', most importantly, 'Finite Fields'. The definitions of these are found in this section. __Group Axioms__
Though not necessarily finite in terms of their order (number of elements.) Groups require fewer axioms to define than rings and fields. They are perhaps more familiar to most people than they would initially think. There are two main types of groups, - Abelian and non-Abelian groups. both of these can be finite or infinite in order.
__Ring And Field Axioms__
Finite fields are fields, and fields are rings. not all rings are fields of course, but fields require stricter definitions (axioms), to distinguish them.
__Subrings, Subfields__
Of course, no structure could claim to have much internal structure were there no way to extend them to higher orders as well as showing examples of internal substructures of lesser order. We find that though some rings have infinite order, so also do their subrings, and mathematically they have the same order (cardinality) although the latter can be properly contained in the former.
__Modular Arithmetic__
In order to discuss finite fields, we see how the basics of modular arithmetic are used. This includes the discussion of equivalence classes, and how the binary operations of addition and multiplication are well defined upon them.
__Ideals Of Rings and Fields__
Ideals of rings are "special" subrings. We see how Fields only have trivial ideals.
__Morphisms / Factor Rings__
Here, things start to get very complicated. We define the kernel of a homomorphism, and state the Ring Isomorphism Theorems. In a parallel of multiplication and division with rings themselves, We briefly touch on the notion of a factor ring.
__Galois Groups And Finite Fields__
Here, automorphisms of finite fields are shown to form a group, moreover one that is cyclic.
__Permutation Groups__
A brief introduction to permutations and their properties. Permutation representation, Symmetric group, Alternating subgroup.
__K4 Groups And The Octal__
The terms "Octal" and "K4 group" are used many times in this website, hopefully after reading this page you should become familiar with these terms. I also define a group product in the octal used extensively in the revelation text which must be understood if a person is to correctly deduce as I have, the correct interpretation of the revelation of Jesus Christ.
__The Finite Fields GF(4) and GF(8)__
These two finite fields are required knowledge for the revelation and metamath sections on this site. You should take the opportunity to become familiar with these mathematical structures before advancing onwards. The structures themselves are well known to mathematicians - fields are frequently used as components in matrices in linear algebra and in vectors, as well as in more general vector spaces and in calculus. You in fact, use fields all the time in algebra: the difference here is that these fields are finite. (They have a finite number of elements.)
__Normal Subgroups__
This page sets the stage for the following pages in the section - greater attention is paid to the concept of a normal subgroup - as it has particular application to the factoring down from S5 to A5 used on this site in numerous places. S5 is the smallest symmetric group to have A5 as a subgroup. A5 itself is the smallest finitely generated non-Abelian simple group. This page simply states why a simple group is insoluble. It occurs in the decomposition series associated with it's normal subgroups, in them not having prime power cyclic factors.
__Simple Groups__
This page uses the complex but powerful machinery of permutations in the symmetric groups to prove a few fundamental theorems concerning normal subgroups of those symmetric groups of degree 5 or more. These theorems provide the basis for the page after this one, which is far more brief and is more a swift summary of the progress made in this page. This proof was taken from my university notes and was taught to me in Autumn 2001.
__A5 Is Simple__
A swift and very brief summation of the proofs in the previous page. By simply assembling the pieces together we now have proof that An is simple for n>4. In particular this includes A5. A5 since it is of index two in S5 is the largest normal subgroup of S5 (and the only one except for C2, its cofactor). This proof is swift but the stages are given here in summary, the previous page is where all the action happened.
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