Prove that for any homomorphism ¤: G → K, ker(@) - G.

Define G: Z →→ Z by G (x) = 5x + 3, for all x € Z. Is G(x) injective, surjective, bijective, or none of these? injective (one-to-one) surjective (onto) bijective [Select] [Select] [Select]

prove that the following two groups of order 6 are isomorphic. Be sure to provide a mapand prove it is an isomorphism. S3 ∼= D3.

(c) Show that if y : G → G' is an isomorphism and : G' → G is a set-theoretic inverse of an isomorphism. then is

Show that if o : G → G' is an isomorphism and : G' → G is a set-theoretic inverse of 9, then v is an isomorphism.

3. whether Gand G'are isomorphic. determine of k 8 G

Consider the set of matrices 0 S = {(a+b):abER} a, bЄR

Let A be an atomic boolean algebra. Let X be the set of atoms. Define the map from A to p(X) as follows a > (a) f is 1-1. (b) ƒ is an isomorphism (c) f is surjective (d) f is not 1-1 {x € X : x < a}.

4. Let p: G → H H be an onto homomorphism. Show that if G is cyclic, so is H.

Show that for distinct i and j from {1,...n} there are exactly two auto- morphisms of Cn which send i to j. Then Show that there are exactly 2n automorphisms of Cn.