Q1 : G = {5, 15, 25, 35} is a group under multiplication modulo 40. Find its left regular representation, i.e., the group of permutations, and express the group element in the regular representation in disjoint cycles.

Q: abouis 13. Let (G, *) be cyclic group of finite order n and let a € G. Prove that ak is a generator…

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

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Q: Find two elements of maximum order in the group G = Z100 Z4 O Z2. How many such elements are there?…

A: As Z100 has no element of order 100 otherwise it will be cyclic which is not true. So, we consider…

Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

A: We know that a finite group G is said to be cyclic if and only if  there exist an element in G such…

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 group of permutations Σ2 is abelian. Then show that Σ3 is not. Writing up the group…

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Q: Let G be a cyclic group of order 135. The number of generators of G is The number of elements of…

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Q: 5. Find the number of generators of the cyclic group Z15

A: To find the number of generators of the cyclic group ℤ15.

Q: [Zp-(0),.] Where p is prime is an abelian group

A: We have to show that [Zp-(0),.] Where p is prime is an abelian group

Q: 6. Let G be a group with the following property: "If a, b, and c belong to G and ab=ca, then b = c."…

A: For a group to be abelian it should follow that ab=ba for all a,b belonging to G. Now we will…

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: Every cyclic group or order n is isomorphic to (Zn, +n) and every infinite cycle group is isomorphic…

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Q: show that if a and b are two different generators of G,* a cyclic group, and therefore abelian, then…

A: If G is cyclic group then order of it's generator is equal to order of group. So, if G has two…

Q: Use the theory of finite abelian groups to find all abelian groups of order 4. Explain why there are…

A: we know that if G is finite group whose order is power of a prime p thenZ(G) has more than one…

Q: 4. List all of the abelian groups of order 24 (up to isomorphism).

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Q: Compute all generators 1) of the multiplicative group Z'n 2) of the multiplicative group Z's.

A: (1)

Q: b. Find all abelian groups, up to isomorphism, of order 720.

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Q: If G is a finite group and some element of G has order equal to the size of G, we ca say that G is:…

A: an abelian group, also called a commutative group, is a group in which the result of applying the…

Q: 6. Embed the group Qs into the SU(2).

A: Given: Q0=e,i,j,k  e-2=e, i2=j2=k2=ijk=e, Where, e is the identity element and e commutes with the…

Q: 10. Prove that any cyclic group is abelian.

A: As you are asked multiple questions but as per our guideline we can solve only one. Please repost…

Q: IfGis a finite group and some element of G has order equal to the size of G, we can say that G is:…

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Q: create the addition and multiplication table in Z5 a) show that (Z5, +) is an abelian group b) show…

A: We know in Z5 addition is defined as,  a+b = (a+b) (mod 5), for all a,b in Z5 In Z5\{0}…

Q: Suppose that G is an Abelian group with an odd number of elements.Show that the product of all of…

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Q: se A cyclic group has a unique generator. f G and G' are both groups then G' nG is a group. A cyclic…

A: 1.False Thus a cyclic group may have more than one generator. However, not all elements of G need be…

Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

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Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

Q: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

Q: 7. Prove that if G is a group of order 1045 and H€ Syl19 (G), K € Syl₁1 (G), then KG a and HC Z(G).…

A: As per policy, we are solving only the first Question, Please post multiple Questions separately.

Q: Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian…

A: To show that S is an abelian group, we have to prove all these properties 1) S is closed under…

Q: Find Aut(Z20). Use the fundamental theorem of Abelian groups to express this group as an external…

A: Find Aut(Z20) by using the fundamental theorem of Abelian groups

Q: ) Describe, up to isomorphism, all abelian groups of order 300 = 22 . 3 - 52 using the Fundamental…

A: Solution: Given : Group of order 300

Q: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

Q: Prove that every finite Abelian group can be expressed as the (external) direct product of cyclic…

A: Fundamental Theorem of Finite Abelian Groups: Every finite Abelian group is a direct product of…

Q: Q2 : Find the left regular representation of the group Z5 and express the group element in the…

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Q: 1. There can only be one group of order h, when h is a prime number (T) or_(F) 2. There is only one…

A: NOTE: According to guideline answer of first three subpart can be given, for other please ask in a…

Q: 2) Prove that Zm × Zn is a cyclic group if and only if gcd(m, n) cyclic group Z; x Z4. = 1. Find all…

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Q: (8) Let n > 2 be an even integer. Show that Dn has at least n/2 subgroups isomorphic to the Klein…

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Q: (а) (Q, +) (b) (Zs, ·)

A: (a)(ℚ,+)(b)(ℤ8,.)

Q: 300Can someone please help me understand the following problem. I need to know how to start the…

A: G is the abelion group of order 16. It is isomorphic to,

Q: Since 11 is an element of the group U(100); it generates a cyclic subgroup Given that 11 has order…

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Q: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

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Q: (d) Prove that 1+ N is a group with respect to multiplication. (e) Verify that 1+N(Z27) is a cyclic…

A: d) To prove that 1+N is a group. Let a=1+n, b=1+n'∈1+N where n,n'∈N=Nℤn. Then,…

Q: Every element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.

Q: Suppose G is a group in which all nonidentity elements have order 2. Prove that G is abelian.

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Q: Determine all invertible elements in Z18 with respect to multiplication. Show that they form a…

A: First we need to find multiplicative inverse of Z18 :

Q: Every abelian cy elic O True O False group is

A: to check  whether every abelian group is cyclic or not? proof let a euler group U8=1,3,5,7 let…

Q: Let m and n be integers that are greater than 1. (a) If m and n are relatively prime, prove that Zm…

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Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

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Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

A: