New York University Non Trivial Group Homomorphism Modern Algebra Exam
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.
- (i) Prove that ‘(g) has finite order in H, and show that the order of ‘(g) divides the order of g.
- (ii) Prove that the order of ‘(g) is equal to the order of g if ‘ is an isomorphism.
- (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?
