Modern Algebra, Groups, Integral domains, fields...

I have an assgnment due on Thursday in Mordern Algebra and I must say that I am very lost I got (I think) 3 problem out of the 6 and I was wondering if someone could help me out and check the work I have done already.

Ok here it is, any suggestion for 1, 2 and 3????

1. Suppose that is a group isomorphism .

Determine

I was trying to find generator of I50 all I found is that and that but doesn’t map all of I50 so I am not sure what to do.

2. Suppose that G is a finite Abelian Group and G has no element of order 2. Show that the mapping is an automorphism of G.

3. Le G be a group of order where p is prime. Prove that the center G cannot have order .

4. Let G be a group. If is a subgroup of G, prove that it is a normal subgroup of G.

My answer: Given is a subgroup of G we must show that

Let and for some .

Since then by definition

Let , with according to the definition of a conjugate.

Then let

By definition , thus .

Therefore H is a normal subgroup.

5.Show that there is an integral domain with exactly 4 elements.

My answer: Let

F4 is a field of 4 elements because it is a commutative ring (close under multiplication and addition), it has a unity and each non zero elements has a multiplicative inverse. Since F4 is a field then it is an integral domain with 4 elements.

6.Let R be an integral domain and . Suppose and . Prove that

My answer:

since .