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

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: 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: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…

Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: No. of isomorphic subgroup of group of integers under addition is: -

A: As we know group of integers under addition is (Z,+)

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: How many elements of a cyclic group with order 10 have order dividing 5?

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Q: Let m be a positive integer. If m is not a prime, prove that the set {1,2,..., m – 1} is not a group…

A: We show that it doesn't satisfy clousre property.

Q: It is known that an algebraic structure, (R, *), is associative, commutative and have identity…

A: Associative property: If a*b*c=a*b*c for all a, b, c∈R where R is an algebraic structure and * is a…

Q: Find any case in which the number of subgroups with an order of 3 can be exactly 4 in the Abelian…

A: Let G be an abelian group of order 108 Find the number of subgroups of order 3. Prove that, in any…

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: How many nonisomorphic abelian groups of order 80000 are there?

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

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Q: Prove that, there is no simple group of order 200.

A: Solution:-

Q: Does the Z base 5 form a cyclic group w.r.t. addition modulo 5? Justify. If yes, find all the…

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Q: Determine all cyclic groups that have exactly two generators.

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Q: 1) (Z,, +,) is a group, [3]- is 2) 11 = 5(mod----) 3) Fis bijective iff

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Q: Is it possible to find a group operation e on a set with 0 elements? With 1 element? Explain why or…

A: The question is :: is there possible to find a group operation on a set of 0  element? Or with 1…

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

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Q: Without using the structure theorem for finite abelian groups, prove that a finite abelian group has…

A: Given: Let G be a finite abelian group. To prove: G has an element of order p, for every prime…

Q: 2. Use one of the Subgroups Tests from Chapter 3 to prove that when G is an Abelian group and when n…

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Q: The operation of composing any two symmetries in a dihedral group is commutative.

A: To find: The given statement true or false. The given statement: "The operation of composing any two…

Q: Construct a subgroup lattice for the group Z/48Z.

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Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: 5: (A) Prove that, every group of prime order is cyclic.

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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: Which abelian somorphic to groups subyraups of Sc. Explin. are

A: Writing a permutation σ∈Sn as a product of n disjoint circles. i.e σ=τ1,τ2,τ3,…τk The order of σ is…

Q: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

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Q: = Prove that, there is no simple group of order 200.

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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: Show that a finite group of even order that has a cyclic Sylow 2-subgroup is not simple

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Q: Show that the multiplicative group Zfi is isomorphic to the additive group Z10.

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Q: Theorem: Any non-commutative group has at least six elements

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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: List six examples of non-Abelian groups of order 24.

A: The Oder is 24

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

A: According to the question,

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: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

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

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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: 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: Show that group Un (n th unit root) and group Zn are isomorphic.

A: There are n elements in the group (Zn,+). There are n elements in the group (Un,×). There are (n!/2)…

Q: Let F denote the set of first fibonacci number .Then convert into a non abelian group by showing all…

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Q: List all abelian groups (up to isomorphism) of order 600

A: To List all abelian groups (up to isomorphism) of order 600

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

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Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

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

A: union of proper subgroups  proof: Let G be a finite group. In the first place, we are going the…

Q: Prove that there is no simple group of order 315 = 32 . 5 . 7.

A: Prove that there is no simple group of order 315=32·5·7.

Q: Let F denote the set of first 8 fibonacci number.Then convert into a non.abelian group by showing…

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Q: Show that a group of order 77 is cyclic.

A: