Problem 1 : G is a group. Define a relation ∼ on G by a ∼ b if a = b or a = b^(−1) . (1) Show that this relation is an equivalence relation. Describe the equivalence classes.

Q: Question 7. Prove or disprove: (a) The union of two subgroups of a group (G, *) is a subgroup of G.…

A: 7)  We need to prove or disprove the given statements. a) The union of two subgroups of a group G ,…

Q: Let P (m, n) be the statement “m divides n," where the domain for both variables consists of all…

A: Since you have posted a multi subparts question according to guildlines I will solve first three…

Q: Qa\ Answer for only one. 1) Let H and K be normal subgroups of a group G, then Hn K be a normal…

A:

Q: 3. Let G be a group and let SG be the set of all permutations on G. For each g € G, define Kg: GG as…

A:

Q: 7. Let G be a group and for elements a, b e G define aRb if and only if there exists an element x E…

A:

Q: Show that isomorphism between groups is an equivalence relation.

A:

Q: Prove that the following two graphs are isomorphic using properties of incidence and adjacency…

A: The star graph with the labels is as follows: The adjacency matrix of the star graph is as follows:…

Q: 7. Let M be the set of all n x n matrices. Define a relation on ~ on M by A B there exists an…

A: A relation ~ is said to be an equivalent relation if it hold following three properties (1)…

Q: Let G be a group and a ∈ G. Suppose that |a| = 18, find |a^21|. Suppose that |a| = 32, find a…

A: We are given that G is a group, and a ∈ G.A) The order of element a is given as 18. a=18 ⇒a18 = e,…

Q: If any two members P and Q of a group satisfy the relation P = Q'PQ, then P and Q commute. Select…

A:

Q: Draw the directed graph of the relation R on A = {1,2,3,4,5,6,7,8} defined as R = {(a,b)|a,b e A and…

A: Here, the relation is: R=(a,b)|a,b∈A and a≡b mod 3 That is: a-b is divisible by 3. So, 7-1=6 is…

Q: is ----Llarder for - 46 8. The Ker() for the homomorphism :Z-Z₁, such that (1)=9is r 6, and r **…

A:

Q: Prove that the following ~ is an equivalence relation: Suppose W = (X,Y) is a graph. We define the…

A: Please see the below picture for detailed solution.

Q: Show that conjugacy is an equivalence relation on a group.

A:

Q: 2. Consider morphisms f: A → B, g: BC, h: CD, k: DB, j: BC, t: A → C in a category C so that g is an…

A: Given: Morphisms f:A→B, g:B→C, h:C→D, k:D→B, j:B→C and l:A→C in a category C, where g is an…

Q: Let G and H be finite groups. Prove the following statements. (a) If a: G H is a homomorphism, then…

A: Given that  and  are finite groups.We need to prove the following.(a) If  is a homomorphism, then…

Q: 5. If the pair of cycles a and ß have no entries in common, then aß= ßa. True False ●

A: answer is in next step

Q: Prove that a kernel of a homomorphism is a subgroup. 52.

A:

Q: Let G be a group. Define a relation on G as follows: N exists an a EG such that y = axa. (In this…

A:

Q: Question 12 Graph Isomorphism: Write an Isomorphism s from G1 to G2, Write your answer as a list,…

A:

Q: 2 20. Let G be a group with subgroup H. Define a relation on G as follows: a b if b-¹a EH. Prove…

A:

Q: 4. Let G = (V, E) and G' = (V', E') be simple graphs. We say that a graph G is homomorphic to a…

A:

Q: Given a group hG, ∗i and a subset H ⊂ G. Prove that H is a subgroup of G if (i) H 6= ∅, and (ii) if…

A:

Q: Let No = (NU{0}) be the set of natural numbers including 0. Let M = N = (No x No × No): relation -…

A: Equivalence relation :                    A relation define on a non empty set S is called an…

Q: 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a)…

A: As per the policy we are solving first question only. Please repost the question and specify which…

Q: Prove that if two transformations are topologically conjugate and one of them mixes topologically,…

A: Given Data: Prove that if two transformations are topologically conjugate and one of them…

Q: Suppose (X,≺) is a well-ordering and f : X → X is an isomorphism. Then f is the identity function on…

A: Well ordering

Q: 10. (1) A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe the…

A:

Q: 1) Let G be a group and H be a subgroup of G then : a = b mod H + b-1 * H = a-1 * H . O true O false…

A:

Q: What is a homomorphism in simplest terms? Also provide an example.

A:

Q: 5. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

Q: Suppose that f:G - G such that f(x) = axa. Then fis a group homomorphism if and only if a^4 = e O…

A: (1) It is given that the function is fx=axa2. For homomorphism, the function follows the condition…

Q: Show that isomophism between group is an equivalence relation. Briefly explain and show all the…

A: Recall the definition of an equivalence relation. It satisfies reflexive, symmetric and transitive.…

Q: It is Possible to define an isomorphism f from the multiplicative group of fourth roots of unity…

A: since for any Isomorphism g from a group G1 to another group G2ge1=e2where e1 and e2 are indentities…

Q: 1. Let G be the set of integers under addition and H = {-1,1} be a multi- plicative group. Show that…

A: