Okay, please let me know if I am on the right track here, any help is very much appreciated. Keep in mind that I am NOT a Math major, so please assume I am not a super wiz on terminology and what not. Thanks.

Let m be a positive integer less than 17. Consider the function f_{m}from Z_{17}to Z_{17}defined by f_{m}(x) = m*x mod 17.

1. Show that f_{m}is one-to-one (and thus a permutation)- This is where I need help the most

2. Determine the disjoint cycle form of f_{7 }

3. Determine if f_{7}is an even permutation

4. Determine the disjoint cycle form of f_{4}

5. Determine if f_{4}is an even permutation

1) So one-to-one means that we must show that it is "injective". Thus, we must show that arbitrary inputs (say a and b) under the same operation in the function must be equal to each other (not sure if I worded this properly)

i.e.

f(x) = x - 5

Let a and b be real numbers such that f(a) = f(b)

f(a) = a - 5 = b - 5 = f(b)

a - 5 = b - 5

a = b

f is injective

So using what know, would we pick three arbitrary values (1 for m and 2 for x)?:

f_{m}(x) = m*x mod 17

Let a, b, and d be arbitrary values such that f_{a}(b) = f_{a}(d)

f_{a}(b) = a*b mod 17 = a*d mod 17 = f_{a}(d)

a*b mod 17 = a*d mod 17

Since 17 is a prime number, it will be relatively prime toano matter the value andais positive (a>0), so we get:

a*b mod 17 = a*d mod 17

-> b mod 17 = d mod 17

I'm not sure where to go from here. I know that the mod function under multiplication in a finite set is closed under the operation and non-repetitive (no two inputs have the same output). But I'm not sure what property to point to or what not to show this. Help here is appreciated.

2 and 3) Here I would show the disjoint cycle form and break it down into disjoint cycles of size 2, and if there are an even amount, it's even, if odd, then it's odd. The only help I need here is can you help me in breaking it up into cycles of 2?

f7(x) = 7x mod 17

So the disjoint cycle form is: (1 7 15 3 4 11 9 12 16 10 2 14 13 6 8 5)

How do I break this into cycles of 2?

Answering the above will help me with the rest. So if you guys could help me out, that would be awesome! Thanks!