FLTMA: A little group theory

Hello. I have completed the first of three tests in the abstract
algebra course at the CoCo. We are up to cyclic subgroups in Fraliegh's
seventh edition of A First Course in Abstract Algebra.

From FLT we have, by way of contradiction (I think that is how you say
it)

a^n + b^n = c^n

and so

x^p + y^p = z^p where gcd(x,y,z)=1, one of {x,y,z} is even, p is prime,
and x<y<z<(x+y).

I'd like to look at (x^p + y^p) (mod z). This implies x^p == -y^p (mod
z). x^p is the additive inverse of y^p, in addition modulo z.

In cyclic group notation in our text we write <x> for all the powers of
x modulo some implied z. <x> is the cyclic subgroup generated by x.

The phi function is written phi(n) = | { x | gcd(x,n)=1, 0<x<n } |.
That is, the measure of the set of numbers coprime to n. We have a
theorem in our text that if element a generates G, that is, <a> = G,
then | <a^s> | = n / (gcd(n,s)). Each of <a^s> is a subgroup of G, and
so contains the inverse of element a^s under the group operation, which
we call *. There are phi(n) generators of an arbitrary group isomorphic
to Zn.

For every z, is there an Abelian group, Zz - 0, of numbers from 1 to
z-1 under multiplication modulo z, with associativity, an identity, 1,
in the group, and a muliplicative inverse for every element in the
group?

If so, then x and y, since they are less than z and nonzero, are
elements of this group, and there are some things that can be said
about them and their powers.

Doug Goncz
Replikon Research
Seven Corners, VA 22044-0394

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