3. Verify Test the properties of a group : Let G denote a set of all ordered pairs of real-valued numbers with nonzero first component. Define the operation * by the rule (a, b) * (c, d) = (ac, be + d), V(a, b), (c, d) E G

Q: Exercise 3.2.6 Show that if G and H are isomorphic groups, then G commutative implies H is…

A: A group G is called Commutative if for any a,b in G imply ab=ba

Q: a. Prove that the set of numbers {1,2, 4,5, 7,8} forms an Abelian group under multiplication modulo…

A:

Q: Suppose G is a cyclic group with an element with infinite order. How many elements of G have finite…

A: Suppose G is a cyclic group with an element with infinite order.  It means that order of group is…

Q: 4. Give an example of a pair of dihedral groups that have the same number of conjugacy classes as…

A: We know that Dihedral group is denoted by Dn. Its order is 2n. Here, to find the pairs of Dihedral…

Q: 6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and…

A:

Q: Is the set of integers a commutative group under the operation of addition? Yes; it satisfies the…

A:

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

A:

Q: 9. Prove that a group of order 3 must be cyclic.

A:

Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: (a) Explain why it is impossible for any set of (real or complex) numbers which contains both 0 and…

A: To solve the given problem, we use the defination of group.

Q: Let G be a group containing elements a and b. Express (ab)^2 without parentheses. Do not assume that…

A: Given that G is a group and let a,b belongs to G. The given expression is

Q: Give an example: The product of two solvable groups need not to be solvable?

A: it is clear that s2 is solvable because it is abelian

Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

Q: Show that the center of a group of order 60 cannot have order 4.

A:

Q: Prove that a group of even order must have an element of order 2.

A:

Q: A group that also satisfies the commutative property is called a(n). (or abelian) group. A group…

A: A group that also satisfies the commutative property is called a(n) commutative group (or abelian)…

Q: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

Q: Prove that a group of order 12 must have an element of order 2.

A:

Q: Find an isomorphism from the group G = to the multiplicative group {1, i, – 1, – i} in Example 3 of…

A:

Q: An element g of G is called idempotent if g2 = g. (1) Show that every group has at least one…

A: For the solution of the problem follow the next steps.

Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

A:

Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

A:

Q: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: The Sylow theorems are significant in the categorization of finite simple groups and are a key…

Q: Compute the center of generalized linear group for n=4

A: To find - Compute the center of generalized linear group for n=4

Q: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

A:

Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

A:

Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

A:

Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

A:

Q: Prove that there is no simple group of order 210 = 2 . 3 . 5 . 7.

A:

Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

A:

Q: iv Sketch the Caley Graph of the additive Group of direct product Z3× Z4 with respect to the…

A: Consider the conditions given in the question. Clearly from the hint a torus is involved in the…

Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

A:

Q: Prove that An even permutation is group w.y.t compostin Compostin function.

A:

Q: (i). There is a simple group of order 2021.

A:

Q: polynomial

A:

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

A:

Q: 2) Determine whether or not the groups Z10 × Z4 and Z, × Z20 are isomorphic. Justify your answer.

A:

Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…

Q: Show that * defined on Z by a*b=|a+b| is not a group. (Hint: identify and show the group property…

A:

Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 7x. Then, Ø-'(7) : O {0, 15, 30, 45} O {1, 6,…

A:

Q: Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the…

A: We know that, Order of the given group is divisible by natural numbers 5,7,8 and 9. So the least…

Q: Exercise 3: Prove that every element of a finite group is of a finite order.

A:

Q: 9. Show that the two groups (R', +) and (R' – {0}, -) are not isomorphic.

A:

Q: Show that a group of order 12 cannot have nine elements of order 2.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: (5) Make a list of all group homomorphisms from Z4 → Z8.

A:

Q: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

A: