3. For each of the following groups G and subsets HCG, determine if H is a subgroup of G and prove your answers. (a) G=C, H = {a+bi | a, b = Z}. (b) G=C, H = {z € C | |z| = 1}. (c) G=C, H = {a + bi | a, b = Z, (a, b) = (0,0)}.

Q: Which of the following groups are solvable groups: Z/nZ, D8, and S4.

A:

Q: 3. Consider the group S. a) For the subgroup H= {(1),(13)} , write all cosets that can be formed. b)…

A:

Q: If you let G and H be two groups and consider the product group G x H and let K be the subset of G x…

A:

Q: Evaluate the expression a+b a+b when a=23a=23 and b=45b=45. Write in simplest form

A: # the expressions are printed twice , taking them once given a=23 and b=45 The value of a+b is as…

Q: Let : G H be a homomorphism. Let (G) = {(g) | ge G}. Show that (G) is a subgroup of H.

A:

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: The following statements are made about (right) cosets of a subgroup H of a group G. Say whether…

A:

Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: x and y are two vertices of a Cayley digraph. ai’s and bj’s are generators of the Cayley digraph

Q: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

Q: Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the group…

A: Given- G=a+b2: a,b∈ℚ and a and b are not both zero To prove- G is a subgroup of ℝ* under the group…

Q: Question 3: a. Let (H,+) ≤ (R, +), let K = {2h : h E H}. Prove that (K,x) ≤ (R*,x). (Clarification:…

A: We shall use subgroup test to solve this problem

Q: 2. Which of the following are groups? For those which are not groups, explain why not (it is enough…

A: 2)(i)  under   is  not  a  group  since which has  no  inverse  under    (ii) Let      Then,   is…

Q: Find all the cosets of the subgroup 4Z of Z

A: In the group of integers under addition, the subgroup consists of all multiples of 4, i.e., .A coset…

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

A:

Q: 1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h)…

A: