User:J D Bowen/Math735 HW1

1.3.16) Suppose $$n \geq m \ $$, and consider the number of m-cycles in $$S_n$$.

An m-cycle is simply a selection of m elements from n without any repeats. The number of permutations of m elements from n possible elements is $$n!/m! = n(n-1)...(n-m+1) \ $$. However, each such string is merely a representation of an m-cycle; the cycle does not depend on the starting element in the string, so we must divide this number by m. Hence, the number of m cycles is

$$\frac{n(n-1)(n-2)... (n-m+1)}{m}$$

1.3.17) Consider a product of two disjoint 2-cycles (wx)(yz), where w,x,y,z are in {1,...,n}. The number of different such arrangements is then n(n-1)(n-2)(n-3). However, this does not account for the following relations:

(wx)(yz)=(xw)(yz)=(wx)(zy)=(xw)(zy)=(yz)(wx)=(yz)(xw)=(zy)(wx)=(zy)(xw)

Since there are 8 different arrangements in the set formed by the n choose 4 operation above, we must divide by 8. Hence, the number of element in $$S_n$$ which are the product of two disjoint 2-cycles is $$n(n-1)(n-2)(n-3)/8$$.

1.4.5) Let $$F \ $$ be a finite field of $$m \ $$ elements. Then the number of $$n\times n \ $$ matrices we can form from this field will be $$mn^2 \ $$, since in each entry of the matrix we have $$m \ $$ choices of element, and there are $$n^2 \ $$ such choices to be made.  This places an upper bound on the number of possible elements in $$GL_n(F) \ $$, since this group will cannot have any element not of this form.

1.4.7) Let $$p \ $$ be prime. Then the general linear group $$GL_2(\mathbb{F}_p) \ $$ will contain all matrices which are invertible, that is, have non-zero determinant.  In order for a 2 by 2 determinant to be zero, one row must be a multiple of another row, and of course there cannot be a row which is all zeroes.

Let us concern ourselves with the first row of an invertible matrix: For the first entry, $$a_{11}$$, we can have any of the p values in the field, and similarly for the second entry $$a_{12}$$. However, they cannot both be zero, or the determinant will be zero. So, there are $$p^2 -1 $$ possibilities for the first row - the $$p^2$$ from all possible options, minus the one option (0,0) which leads to a null determinant.

For the second row, we cannot have (0,0) either, or any multiple of the first row. Consider that (0,0) IS a multiple of the first row, if we multiply by 0. Hence, for the second row, there are $$p^2-p$$ possibilites - $$p^2$$ from the total possibilities, minus the p possibilities which are multiples of the first row.

Hence, the number of invertible matrices is $$(p^2-1)(p^2-p)=p^4-p^3-p^2+p \ $$.