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: Show that if H and K are subgroups of G then so is H ∩ K.

A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…

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

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Q: 15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk =…

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

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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…

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Q: Suppose o is a homomorphism of groups fom G onto H. If H has an element of order 100, then show that…

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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: (3) Let (A, +..) be a subgroup of (M₂ (Z), +,.), Then A is ideal of M₂ (Z), where A = {(a b) la, b,…

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

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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: 7. Let G be an infinite cyclic group. Give an example of a subgroup H in G such that H {e} or G. 8.…

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Q: 1. Consider the following groups G and H, K < G. If G is isomorphic to H × K, give an isomorphism v…

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

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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: Q5. Let A and B be two groups. Let 0: A x B → B defined by 0(a, b) = b Is 0 isomorphism? Find…

A: To check whether a function θ is isomorphism, it is required to check  θ is homorphism θ is one-one…

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: 5. If H = 122Z and K = 8Z are subgroups of (Z, +). Then H + K = ... %3D

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Q: Find all the subgroups of Z48. Then draw its lattice of subgroups diagram.

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

A: It is given that x and y be two vertices of a Cayley digraph. We can write two paths as a1,a2,a3,…an…

Q: 1. Let (G, *) be a finite group of order n. Then (a). Let n = pq (p, q are a prime numbers) and let…

A: This is a problem of Group Theory.

Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

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Q: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

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

A: Let H be a subgroup of S5. we see that, HA5=HA5H∩A5 It follows that, A5≤HA5H∩A5≤S5 Since, S5=2A5, so…

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: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

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Q: Find the number of G-orbits in X := {1,2,3, 4, 5, 6, 7, 8, 9, 10}, where G < S10 is the subgroup…

A: Solution:Let g1 = (1,2), g2 = (3,4,5) - and g3 = (6,7,8,9). All three cycles at sight are…

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: Let G be a finite group and H1, H2, ., Hk be subgroups of G. (a) Show that k H1n H2n. n Hk < G. Hi…

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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: 4. Let G and G be groups and let y be an isomorphism from G to G. Prove that if KCG, then y(K) CG.

A:

Q: ACR group A = {x+y: −1 < x≤0, 2 < y ≤ 3}. Define group A's infinum (inf A).

A: As  -1 &lt; x And  2 &lt; y Then   -1+2 &lt; x+y I. e. 1 &lt; x+y

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

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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)…

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