What is the smallest positive integer n such that there are two nonisomorphicgroups of order n? Name the two groups.

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

Q: When do you use a t-test for independent groups and what do the numerator and deniminator in the…

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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: Prove that every group of order 330 is not simple.

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Q: 2. Are the groups (R, +) and (R†,') isomorphic? Justify your answer.

A:

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: In the group of integers Z with addition, cl(5) the conjugation class of 5 is equal to:

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

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

Q: 25. o: Z4 → Z12

A: Homomorphism : Let us consider a map f: V→W then f is said to be homomorphism if for all v,u∈V…

Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: Explain why the only simple, cyclic groups are those of prime order.

A: Proof: Let G be a simple group with |G|>1. We want to prove that G is a cyclic group of prime…

Q: Let p1, p2 ......pn be distinct primes. Up to isomorphism, how manyAbelian groups are there of order…

A: Given Data The distinct prime is p1, p2…………. pn.   The number of Abelian groups of the order pm, so…

Q: How many elements of order 5 might be contained in a group of order 20?

A: using  third Sylow Theorem

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

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Q: Every group of order 4 is cyclic. True or False then why

A: Solution

Q: (a) What does it mean for two groups to be isomorphic?

A: see my solution below

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

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Q: Z, x is not a group.

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Q: How many groups are there of order 12 (up to isomorphism) 5 O None of them 2 O 4 O 8 0

A: Use the concept of isomorphism.

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.

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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: Show that there are no simple groups of order 255 = (3)(5)(17).

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Q: Let p be a prime number and (G, *) a finite group IGI= p?. How can you prove that the group (G, *)…

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Q: True or False? The only simple groups are the groups Zp and An  where p is a prime and n≠4 .

A: False.

Q: c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

A: #Dear user there is a mistake in the question the assumption is for the element c and d of a group…

Q: Let φ : Z2 → Z9 be defined by φ(n) = n (mod 2). Is φ a group homomorphism?

A: Let φ:ℤ2→ℤ9 defined by φ(n)=n(mod 2) We have ℤ2=0¯,1¯, ℤ9=0¯,1¯,...,8¯

Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: Show that any group of order less than 60 is cyclic

A: This result is not correct. There is a group of order less than 60 which is not cyclic.

Q: If G is an infinite group, what can you say about the number ofelements of order 8 in the group?…

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Q: 14. Prove that the set of all rational number of the form 3"6" | m,nEZ} js a group under…

A: Denote with

Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

Q: Characterize those integers n such that the only Abelian groups oforder n are cyclic.

A: According to the question,

Q: Prove that there are exactly five groups with eight elements, up to isomorphism.

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Q: What limits will include the middle 80% of the group?

A: The required calculation can be defined by the formula Z=x-μσP(a≤Z≤b)=P(b≤Z)-P(a≤Z) Given, What…

Q: Which of the following is nontrivial proper sub- group of Z4? {0, 2} O Diophantus of Alexandria {0,…

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Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

Q: Verify the corollary to the Fundamental Theorem of FiniteAbelian Groups in the case that the group…

A: To verify corollary to the Fundamental Theorem Of Finite Abelian Groups Where, G is a group of order…

Q: What is the smallest positive integer n such that there are exactlyfour nonisomorphic Abelian groups…

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Q: True or False: (a) Two finite non-cyclic groups are isomorphic if they have the same order. (b)Let o…

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Q: Show that there is no simple group G of order 33, 34, 35, 38, 39, 42, 44, 46, 50, 51. Show that…

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Q: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

Q: True or false? Every group of 125 elements has at least 5 elements that commute with every element…

A: Let G be a group whose order is 125 ⇒G=125=53 Center of a group G ( ZG ) is the set of all those…

Q: 4. Consider the additive group Z. Z Prove that nZ Zn for any neZ+.

A: We know that a group G is said to a cyclic group if there exists an element x of the group G such…

Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

Q: What is the numbers group of

A: The given number is 5. The value of 5 is 2.23606...

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: 2) Given example of an infinite group in which every nontrivial subgroup is infinite.

A: Let G=a be an infinite cyclic group generated by a, whose identity element is e. Let g∈G, g≠e,…

Q: 5. Let a be an element of order n in a group and let k be a positive integer. Then =< a™dlnA)

A: To prove : ak=agcd(n,k) Let set d = gcd(n,k)  and then write k=dr by definition of gcd, We prove…