I am learning abstract algebra. Sorry to put so many questions here. Look forward to your help.Thanks a million

 

1.Prove any group of order 765 is abelian

Prove that a group of order 135 has a normal subgroup of order 15

 

2.Let A be an abelian group with no elements of infinite order.

Suppose that every element of prime order is of order 3.

Show that the order of every element is a power of 3.

 

3.If G is finite and G/(A^B) isomorphic to (G/A) X (G/B) , prove that AB= G

(A^B means the intersection of A and B)

 

4.If S is a simple nonabelian group, prove that contains a subgroup isomorphic to S

 

5.Give an example of fields such that K a normal extension of F and L a normal extension of K, but L is not a normal extension of F

 

6.Explain how one determines the number of ideals of Z/(n) in terms of n,where n is a positive integer

 

7.Let K be a field of characteristic not equal to 2 . Suppose f(x) = p(x)/q(x) is a ratio of polynomials in K[x]. Prove that if f(x) = f(-x), then there are polynomials p0(x^2), q0(x^2) such that f(x) = p0(x^2)/q0(x^2).

 

Appreciate your help!

 

Shelley

The answers all follow with a littel thught and care about the definitions,. Try and think about them a little harder. Though 4 makes no sense. 2 is very easy: if a is of order pq then a^q is of order..... 5 is standard and can be found in any galois theory text., for 6 one needs only to mention the word euler and possibly totient...