Abstract algebra. proving things about U(n)

[cap.r]

Homework Statement

prove that if g is in Z*_n then g^2=1, so g has order 2 or is the identity.
show that the largest value of n for which every non identity element of Z*_n has order 2. which are these others.

Homework Equations

Z*_n = U(n) different notation it is the the group of co primes to n less than n.
For example: Z*_10 =U(10)={1,3,7,9}, with 3x7=1; 7x7=9 etc.

The Attempt at a Solution

ok the problem as 3 parts.
1) Show that for every element g in U(24), g^2 = 1.
This is easy to check by direct computation. e.g. 5x5=1, 11x11=1 etc.

2) Find all m < 24, such that every g in U(m)has the property g^2=1(mod m).
My hunch is: we must have m = 1, 2, 3, 4, 6, 8, 12
I have to show that the above statement is correct and then also
establish that for other m <24
(i.e m=5, 7, 9, 10,...,23) there is some g in U(m) for which
g^2 is *NOT* equal to 1(mod m). I need help with this part.

3) The third part is to show that:
for all m > 24, there is a g in U(m) such that
g^2 is *NOT* equal to 1(mod m).
dont' know how to do this part either. but i think getting part 2 will help with this part.

 

[cap.r]

just bumping this hoping to get more views. I am still stuck and not sure how to proceed.

 

[cap.r]

ok so I have noticed that the only numbers that posses this property are number where U(n) is a set of only primes. for example U(9) includes 4. and 2^2 is not 1 mod 9. but the numbers 12,8,6,4,2 all have the property that their U(n) ring contains only prime numbers and thus they posses the desired property. but I don't know how to prove this.

also I need to show that the U(n) for any number greater than 24 always contains a non prime number. i feel like i can do this, but the fact that all the elements in U(n) must be prime is what still needs to be proven.