S_n has order n!

The order of S_n is the number of permutations of n elements? So, you have n blank spaces and can choose from n different elements to fill the first space, from the remaining n-1 to fill the 2nd space, etc. until you're at the last space and only have one element left. Using the multiplication rule, that gives n(n-1)(n-2)(n-3)...1 = n! different permutations.

How are you getting the grey boxes to write math?

The even integers is a subgroup of the integers

Let a and b be even integers. Showing that a - b is also an even integer suffices, by your criterion above.
Factoring out twos: a= 2n and b = 2m for some integers n and m, so a - b = 2n - 2m = 2(n - m), and n - m is an integer. So a - b is even.