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 . ...continues

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. Sh ...continues

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.

Definitions of Equivalence & Groups

(1) State the definition of equivalence relation.... and (2) Give one example of an abelian group and two (2) examples of nonabelian groups

Group Homomorphism

Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi). proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel, but I'm not sure about how to do the reverse (<=) an ...continues

Symmetric Group Question

It is trivial that S[n] is cyclic for n = 1, 2, but is S[n] ever cyclic for n>=3? Prove why or why not.

Normal Subgroups

Find all maximal normal subgroups of Z[p] × Z[q], where p and q are relatively prime. Would the elements from Z[p] have to be one that are relatively prime to q and vice versa?

Group of Permutations

Consider the group Z[4] × Z[6] under * such that (a, b) * (c, d) = (a +[4] c, b +[6] d). (here +[4] means + is in Z[4] and +[6] is in Z[6]) We would like to find a group of permutations that is isomorphic to Z[4]Z[6]. Is this group cyclic? If so, prove it. If not, explain why. Do I need to list all the members and che ...continues