total of 9 questions please read the details

1. 2.

Let n 2 N, let G be a group of order 2n, and let H be a subgroup of order n. Prove that gH = Hg for all g 2 G.

(i) List all the permutations in A4. (ii) Let

H = {1, (12)(34), (13)(24), (14)(23)}
be a subgroup of A4. Find the left cosets of H, and the right cosets of H.

(iii) Using your results from part (ii), decide if H is a normal subgroup of A4. (iv) Let

K = {1, (234), (243)}
be a subgroup of A4. Find the left cosets of K, and the right cosets of K.

(v) Using your results from part (iv), decide if K is a normal subgroup of A4. (vi) Calculate [A4 : H] and [A4 : K].

Let ': G ! H be a group homomorphism. Let g 2 G be an element of finite order.

1. (i) Prove that '(g) has finite order in H, and show that the order of '(g) divides the order of g.
2. (ii) Prove that the order of '(g) is equal to the order of g if ' is an isomorphism.
3. (iii) By considering elements of order 2, explain why D6 is not isomorphic to A4.

Let ': G ! H be a non-trivial group homomorphism. Suppose that |G| = 42 and |H| = 35.

(i) What is the order of ker '?
(ii) What is the order of the image of '?

(i) Solve the congruence 7x ⌘ 13 mod 11. (ii) Solve the equation 6x = 17 in F19.

(iii) How many solutions does the equation 6x = 5 have in Z/9Z?