Congruences, Equivalence Relations and Inverses

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

Set Theory and Operations and Surjective but not Injective Mappings

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, U = union (2) (A - B) U C = (3) Give an example that a mapping from A to B that is surjective but not injective.

Factor Groups of Non-Abelian Groups

Let G be a nonabelian 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?

Definitions and Examples : Groups, Abelian Groups and Non-Abelian Groups

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 and Abelian Groups

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 Groups

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?

Isomorphisms, Cyclic Groups and Groups 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

Normal Subgroups

Let G be a finite group, let N be a normal subgroup of G, and let x be an element of G. Show that if the order of x in G is relatively prime to |G|/|N|, then x is an element of N. We know that xNx^(-1) is identical to N when N is normal, for any x. Also we know that |G|/|N| is a factor of (or divides) |G|. How to show x i ...continues