Abstract Algebra

First off, I'll explain what axioms are since they are of fundamental importance to abstract algebra and math in general. In mathematical logic, an axiom is an underivable, unprovable statement that is accepted to be truth. Axioms are, therefore, statements which form the mathematical basis from which all other theorems can be derived. The most well-known modern examples of mathematical axioms are those of the axiomatic set theory known as ZFC.

The Zermelo-Fraenkel set theory with the axiom of choice added to it, abbreviated ZFC, is the axiomatic set theory which provides our basis for math. There are nine axioms in the theory:

1) Two sets are equal if and only if they have the exact same members.

2) A set can't be an element of itself.

3) If there is a property that is characteristic of the elements of a set, a subset of that set exists containing the elements that satisfy the property.

4) A set exists containing exactly all of the members of two given sets.

5) A set exists whose elements are the members of the members (the union) of a given set.

6) The image of any function on a set is also a set.

7) There is a set that contains all of the natural numbers.

8) Every set has a power set; i.e. the set of all possible subsets of the original set.

9) Every set can be well-ordered such that every subset of the set has a "least" element under the ordering.

All nit-picking aside, these nine axioms are mathematically unprovable and therefore must be assumed true for mathematics to work.

A conjecture, as opposed to an axiom, is an unproved (not unprovable) statement that is also generally accepted to be true. The subtle difference between the two terms is basically that an axiom has been proven to be unprovable, whereas a conjecture hasn't.

See the related link for more information about ZFC.

Say we have a group G, and some subgroup H. The number of cosets of H in G is called the index of H in G. This is written [G:H].

If G and H are finite, [G:H] is just |G|/|H|.

What if they are infinite? Here is an example. Let G be the integers under addition. Let H be the even integers under addition, a subgroup. The cosets of H in G are H and H+1. H+1 is the set of all even integers + 1, so the set of all odd integers. Here we have partitioned the integers into two cosets, even and odd integers. So [G:H] is 2.

A matrix is symmetric when it is a perfect square and the transpose of the matrix is equal to the original matrix. The implementation is reasonably straightforward and can be done without actually creating the transposed matrix. We simply imagine a dividing line through the main diagonal (top-left to bottom right) and compare the corresponding elements on each side of that diagonal.

bool is_symmetric (int* a, int rows, int cols) {

if (rows!=cols) return false;

for (int row=0, row

for (int col=row+1; col

if (a[row][col] != a[col][row]) return false;

return true;

}

This works because for every element a[row][col] in a square matrix, the corresponding element on the other side of the main diagonal must be a[col][row]. In other words, we simply transpose the subscripts (hence it is called the transpose).

Consider a 4x4 matrix of hexadecimal values:

0 1 2 3

4 5 6 7

8 9 A B

C D E F

The main diagonal line is identified as elements {0, 5, A, F}. Element {7} lies above this line and it corresponds with element {D} below the line. Element {7} is identified with the subscript [1][3] while element {D} is identified with the subscript [3][1].

Note that the inner loop which traverses the columns always starts at col=row+1. This ensures that we always start with the first element after the diagonal element in the current row, and traverse through the remaining elements of that row. In other words, elements in [row][col] are always in the upper-right of the matrix, above the main diagonal. Meanwhile, the corresponding transpose, element [col][row], lies in the lower-left of the matrix, below the main diagonal. There is no need to test the entire array, traversing every column of every row, because there's no point in testing elements on the diagonal itself (we'd be comparing each element to itself and its given that an element is always equal to itself), and there's no point in comparing the lower-left elements to the upper-right elements when we've already performed the reverse comparison. After all, if X==Y then it follows that Y==X because equality is a transitive operation (the result is the same regardless of which side of the operator we place the operands).

Note that in order to replace a square matrix with its transpose, we simply swap the corresponding elements unless the matrix is symmetrical (because, by definition, a symmetrical matrix is equal to its own transpose):

void transpose (int* a, int rows, int cols) {

if (rows!=cols is_symmetric (a, rows, cols)) return;

for (int row=0, row

for (int col=row+1; col

swap (&a[row][col], &a[col][row]);

}