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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.