Counter examples, Group and Ring theory

[Meital]

Hello all..I am taking abstract algebra class and I am practicing problems dealing with ring theory. One of the problems I have is not being able to construct examples or provide counter examples. For some reasons, most of the counter examples turn out to be either S_3 or group of quaternions. Are there famous groups that one needs to know in order to be able to construct counter examples to famous theorems in group and ring theory? Also, I wonder if someone knows an example of an integral domain which has an infinite number of elements, yet is of finite characteristic?

[matt grime]

S_3 is the smallest non-abelian group, that is why it is used - it will often be the simplest counter example. And Q is the smallest non-abelian p-group. Hence it's use.

 

Others are: A_5 smallest simple group. S_6 only permutation group to possess an outer automorphism.

 

Yes, there are infinite fields of non-zero characteristic - the algebraic closure of F_p

[Meital]

Is F_p the set of integers that are modulo p, given that p is some prime number?

[matt grime]

Yep, that's what F_p is.

 

Another way of creating an infinte field of characteristic p is to take (p), 'the' maximal ideal in the ring of algebraic integers, A, containing p, and let F= A/(p).

[matt grime]

Here are some other important facts that show why S_3 amd Q and A5 are important.

 

p, q, r will be distinct primes.

Every group of order p is cyclic - hence C2xC2 is the first possible non-cyclic group.

Every group of order p^2 is abelian, hence S_3 is the first possible non-abelian group.

Combining these two, Q is the first possible non-abelian p-group.

Every group of order p^a.q^b is solvable. (Burnside)

Every group of order pqr is solvable.

Hence 60 is the smallest possible order for a simple group. A5 is simple and of order 60.

 

What is amazing is that the smallest order for a counter example actually has a counter example. Well, I think that's quite astounding.

 

Q gives us the quaternionic algebra. This is the first division algebra that isn't a field you will meet, too.