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Regarding group conditions, all books say closure, associativity, same identity (left-side, right-side i.e a*e = e*a =a) are required. But about inverse they differ. Some books say, for every element a, there should be an element such b that a*b = e where e is identity element. But some other say that for every element a there should be such that a*b =e. Without mentioning that b*a also should be e,. b*a = e is it not required. Can it be derived from other properties?
I know how to build GF(p) where p is a prime. It is just simply {0,1,2, ....p-1) with modulo addition and modulo multiplication p operations. From the number theory, we can show that the set really satisfies the 11 requirements of a field iff p is a prime. I also know that similar things happen if we take irreducible polynomials of degree any natural number m over a field - if we take addition and multiplication modulo this polynomial. So we can take an irreducible polynomial and build a GF(p^m). We can build the Cayley's tables for these two operations. Using the tables, we can find the orders of each of the elements and find the primitive roots. But error control coding books follow a different approach. They use primitive polynomials and build the field from the root of this 'primitive polynomial'. I do not know the advantage of following this approach. And actually I am confused with this approach. For example, assuming a root for an irreducible polynomial to start with, whereas root does not exist in the field. For example existence of root for 1+X+X^3 while considering a binary field. Can some one help understand the rationale of this approach?
Dear Bob Bundy,
Thanks a lot for the clear proof of impossibility of union and intersection as the operation of a group. But there can be other set operations like difference, symmetric difference, complement of sets etc - any thing based on the elements of the sets . Can we show that they are all ruled out?
Thanks,
Seetha Rama Raju Sanapala
I want to know if we can have a group of at least 2 elements with sets as elements and some set operation (union, intersection or some other) as the binary operation.
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