Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group itself). Generalize to exactly nsubgroups for any positive integer n.

Q: G is a group

A: Let G={0,♥}Define operation + as follows in operation table+0♥00♥♥♥0then + is a binary operation

Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

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

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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: What is the smallest positive integer n such that there are three nonisomorphicAbelian groups of…

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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: Show that if n > 6, then the symmetric group Sn contains an element of order greater than n

A: According to the problem, we have

Q: How many proper subgroups are there in a cyclic group of order 12?

A: let G be a group of order 12 and let x be the generator of the group. Then the group generated by x,…

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: Remark: If (H, ) and (K,) are subgroup of a group (G, ) there fore (HUK, ) need not be a subgroup of…

A: Definition of subgroup: Let (G ,*) be a group and H be a subset of G then H is said to be subgroup…

Q: How many subgroups of a not abelian group of order 6 is non-cyclic? Select one:

A: Given: The order of the group = 6.

Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of the order…

A: Here given that G is a finite Abelian group that has exactly one subgroup for each divisor of the…

Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: How many elements of a cyclic group with order 14 have order 7?

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Q: If N is a subgroup of an Abelian group, prove that is Abelian. N |

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Q: Show that a subgroup of a solvable group is solvable.

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Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is 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: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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Q: Give a list of all groups of order 8 and show why they are not isomorphic. for this you can show…

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Q: List all elements of the group U(15). Is this group cyclic?

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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: Suppose that G is a cyclic group and that 6 divides |G|. How manyelements of order 6 does G have? If…

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Q: Suppose that G is a finite group and that Z10 is a homomorphicimage of G. What can we say about |G|?…

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Q: The order of identity element equal itself in any group

A: .

Q: Give an example of subgroups H and K of a group G such that HKis not a subgroup of G.

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Q: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

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: (a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…

A: PART A: Since, Every subgroup is a group in itself. To prove <g> is a subgroup ,One step…

Q: What is the smallest positive integer n such that there are two nonisomorphicgroups of order n? Name…

A: Non-isomorphic groups: Groups that have different Sylow-2 groups are non-isomorphic groups.

Q: 8. Give an example of a group G where the set of all elements that are their own inverses does NOT…

A: Let, G,. is a group. Let, G={1,7,17,2,12,3,13} Let, H be a subgroup of G where H={1,7,17,2,12}

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

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Q: Prove that every subgroup of nilpotent group is nilpotent

A: Consider the provided question, We know that, prove that every subgroup of nilpotent group is…

Q: Every finite group is cyclic

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Q: Let (G,*) be a finite group of prime order then (G,*) is a cyclic

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Q: If H and K are subgroups of G, Show tht H intersecting with K is a subgroup of G.  (Can you see that…

A: Use the 2-step subgroup test to prove H Ո K is a subgroup, which states that,

Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…

A: The identity element in a subgroup H of a group G must be the same as the identity element in G.

Q: Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G…

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Q: Find an example of a noncyclic group, all of whose proper subgroupsare cyclic.

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Q: The Kernal of any group homomorphism is normal subgroup True False

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Q: a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…

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Q: Check whether a group of order 156 is simple or not.

A: Definition: A group G is said to be simple if the only normal subgroups of G are the trivial group e…

Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N

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Q: Give the Cayley table for the group Z2 under multiplication modulo 12.

A: Since , We know that Z12 = 0,1,2,3,4,5,6,7,8,9,10,11 and…

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: 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: i have included a picture of the question i need help understanding.thank you in advance. please…

A: Let H and K are two subgroups of the group G.To show

Q: Consider the alternating group A4. Identify the groups N and A4 /N up to an isomorphism.

A: Consider the alternating group A4. We need to Identify the groups N and A4 /N up to an isomorphism.…

Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution

Q: Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element.…

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