Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to Z12 and a subgroup isomorphic to Z20.  No need to prove anything, but explain your reasoning

Q: Write ?(16) as an internal direct product of its subgroups.

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Q: Describe the factor group Describe the fa ctor group. (z/(u).)

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Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: a.)Explain what’s special about a cyclic subgroup. b.)Why would all of the “powers” (don’t forget…

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Q: Applying what we discussed in cyclic groups, draw the subgroup lattice diagram for Z36 and U(12).

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

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Q: example of a non-cyolie group but all group which are cyclic Give an

A: The group U(8) = {1, 3, 5, 7} is noncyclic since 11 = 32 = 52 = 72 = 1 (so there areno generators).…

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: Let G be a cyclic group and H be a subgroup of G. Prove that H is a normal subgroup in G and G/H is…

A: Given:- H be a normal subgroup of a group  G⇒gHg-1=H∀g∈G Also given GH  is a cyclic and we need to…

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 G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

Q: Let G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Given that G is a group and H, K are subgroups of G with the condition that H=<a> and…

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: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

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Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

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Q: Classify groups of order 2p as best as you can. Give a proof for your assertions and, when…

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Q: Please help me understand the following question and please explain the steps. Picture is below

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Q: True or false? The union of any two subgroups of a group G is a subgroup of G.

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Q: Let Z denote the group of integers under addition. Is every subgroupof Z cyclic? Why? Describe all…

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Q: What is a quotient group and conjugacy class

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Q: Is the set of numbers described below a group under the given operation? Natural numbers;…

A: Given is a statement. is the set of Natural numbers is group under subtraction?

Q: B. Let G be a group of order 60. Is there exist a subgroup of G of order 24? Explain your answer

A: We have to give reason that is it possible a subgroup of order 24 of a group of order 60.

Q: Give an example or explain why the following is not possible.  An infinite group that is finitely…

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Q: Give an example, with justification, of a semi-group which is not a group.

A: A semigroup defined on a set is a binary operator which satisfies the association property of…

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: Find the Sylow-subgroups of group G= Z0. Write as direct sum.

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Q: Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic

A: The proof has if part and only if part of the proof. If part: We are given that  a finite group is…

Q: Modern Algebra: Give a counter-example to the statement (1) for a non-normal subgroup N. (1) Let P…

A: Given, P be a Sylow p-subgroup of a finite group G and N is a normal subgroup of G.  To prove:  P∩N…

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

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Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

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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: (a) How Can we tind thał the groups Give three points and two two not ISomorphic, are examples:-

A: Three points that can be used to prove that two groups are not isomorphic are: Cardinality of the…

Q: polynomial

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Q: Find a set of generators and relations for Q8

A: Given: The Quaternion Group Q8. To find: The set of generators and relations.

Q: 26. Show that any finite subgroup of the multiplicative group of a field is cyclic.

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Q: Is the set of numbers described below a group under the given operation? Even integers; addition O…

A: A set is called group on binary operation if a) It is closed b) It is associative c) It has identity…

Q: Is the set of numbers described below a group under the given operation? Rational numbers;…

A: This is a question of abstract algebra.

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

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Q: What can you say about the centre of a simple group?

A: 9

Q: The centralizer and normalizer of a subset of a group are same . its true give proof if its not true…

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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: What is the numbers group of

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

Q: If possible, give two examples of nontrivial proper subgroups of the indicated group. Ifit is not…

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Q: a) Clearly state all the subgroups of the additive group Explain why/how you have found all the…

A: We have to write subgroup of z_9.

Q: Is the set of numbers described below a group under the given operation? Integers; addition Yes O No

A: Given: A set of integers under the addition operation. We have determine whether the set of integers…

Q: et G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

A: Since H is a cyclic group. and K is cyclic group. H=a,K=b If H⊂K then H∩K=H where H is cyclic If K⊂H…

Q: 21. Define Alternating group. Can you explain this concept (order, is it subgroup of Sn)?

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

A: We need to show that (G,*) is cyclic.

Q: Give an example of a finite group that is not abelian.

A: We have to give an example of a finite group that is not abelian. Pre-requisite : A non-abelian…

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

A: Need to find intersection of subgroup