Number theory and algebra theory?

Question

1.Prove that a decimal (ie base 10) number a(subscript n)…a(subscript 2), a(subscript 1), a(subscript 0) is divisible by 11 if and only if the number (-1)^n*( a(subscript n)+…+a(subscript 2) - a(subscript 1) + a(subscript 0) is divisible by 11.

2.Evaluate (59/61), (30/59) showing all the steps and naming the laws used at each step. (Hint: Legendre’s symbols.)

3.Prove that if G=, then G=

4. Let G be a finite group, and let g,x∈ G
(a) prove that x and g^-1xg have the same order
(b) Let a,b∈G. Prove that ab and ba have the same order

This is i think the last of my exam help..thanks to everyone. i really appreciate it

Don

Answer 1

I answered #1 in another thread. The core idea is that 10 is congruent to -1 modulo 11, and hence every power of 10 is congruent to 1 or -1.

For #3, let the order of x be k. Then x^(k-1) = x^(-1), and also (x-inverse)^(k-1) = x. That's enough to show inclusion between the groups both ways, and hence equality.

For 4a, if one of the two terms to the kth power equals 1, so does the other one. To see that, just raise the conjugate to the kth power and cancel what you can cancel.

It's not clear what #2 means.