Group Theory Questions

1. Let f : X -> Y and g : Y -> Z be mappings. (1) Show that if f and g are both injective, then so is g o f : X -> Z (2) Show that if f and g are both surjective, then so is g o f : X -> Z. 2. Let alpha = 1 2 3 4 5 and Beta = 1 2 3 4 5 3 5 1 2 4 3 2 4 5 1 . (I couldn't draw ( ) on both sides of these groups. (1) Write Alpha and Beta into cycles. (2) Find AlphaBeta and BetaAlpha. Is AlphaBeta = BetaAlpha? (3) FInd Alpha^2 , Beta^2 and(AlphaBeta)^2 (4) Find Alpha^2Beta^2. Is Alpha^2Beta^2 = (Alpha-Beta)^2? (5) Fnd Alpha^-1 and Beta^-1. 3. Let Alpha, Beta E S_n . Show that if AlphaBeta = BetaAlpha, then ... click for more

Show that a = b mod m is an equivalence relation on Z.

1. Show that a = b mod m is an equivalence relation on Z. I used = to mean "equal by definition to" and Z as integers. 2. Find the inverse of each of the following integers. r 1 2 3 4 5 6 ----------------------------------- r^-1 mod 7 3. Show that there are no integers x and y such that x^2 and Y^2 = 19. I think with these examples I can figure out some of the others.

Let A={-1,0,1,2} , B = {-2,3,4} and C= {-2,0,1,4}.

Let A={-1,0,1,2} , B = {-2,3,4} and C= {-2,0,1,4}. Find: (1) (A U B) ^ C = I used ^ for "intersected with" symbol (2) (A - B) U C = (3) Give an example that a mapping from A to B that is surjective but not injective.

Factor Group of a non-abelian group

Let G be a non-abelian group and Z(G) be its center. Show that the factor group G/Z(G) is not a cyclic group. We know if G is abelian, Z(G)=G. But now if it is not abelian, can we simply say because G is not cyclic, then any factor group will not be cyclic either? or is there more to it?

Definition of Group

1. I need a simple definition of a (1) group (2) abelian group (3) nonabelian group 2. Give one example of an abelian group and 2 examples of nonabelian groups.