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

Q: Which of the following are subgroups of the group Z×11: {1,3,4,5,9}, {1,3,5,7,8}, {1,8},  {1,10},…

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Q: An is the alternating group on n letters. What is |An|?

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Q: Let G be the symmetry group of a circle. Explain why G containsDn for all n.

A: Given: G is the symmetry group of a circle.We need to explain that G contains Dn for all n.

Q: Consider the dihedral group D6.Find the subgroups of order 2,3,4 and 6

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Q: Suppose G is a cyclic group with an element with infinite order. How many elements of G have finite…

A: Suppose G is a cyclic group with an element with infinite order.  It means that order of group is…

Q: Let 3 denote the centre of a group G. then show that a 3 iff N(a) = G. also show that if G is…

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Q: Give examples of four groups of order 12, no two of which are isomorphic.Give reasons why no two are…

A: We have to give 4 examples of order 12, such that all are non-isomorphic to each other.

Q: Find all the conjugate subgroups of S3, which are conjugate to C2 .

A: Given-S3 To find- all the conjugate subgroup of S3 which are conjugate

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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Q: Suppose T: G-> H is group isomorphism and S:H-> K is also a group isomorphism. SHow the…

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Q: Find all subgroups of the dihedral group D3 with a detailed explanation.

A: Find all subgroups of the dihedral group D3 with a detailed explanation.

Q: . Consider the group (Z30, +). (a) Find the orders of the elements 3, 5, 6, 7,8, 18, 19, 24 and 25.…

A: The order of an element is n , when n is the least such that, n.a=0 3.10 = 0 and 19 the least , so…

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: 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: b. Find all abelian groups, up to isomorphism, of order 360.

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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: Suppose that a group contains elements of orders 1 through 10.What is the minimum possible order of…

A: Given that, a group contains elements of orders 1 through 10.We have to find the minimum possible…

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

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Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4.

A: It is given that a Sylow 2-subgroup of S4. So, we have, S4=4!=24=3×23. let k be a Sylow 2-subgroup…

Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A: Given set is G = <a, b | a^6 = b^3 = e, b^-1ab = a^3>.

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

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Q: What is a quotient group and conjugacy class

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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: write why, in order to describe the homomorphic images of a group, it suffices to describe the…

A: The homomorphic images of a group refer to the set of all possible images that result from applying…

Q: Show that (up to isomorphism) A5 is the only simple group of order60.

A: Given : G=60 To Prove : G≃A5

Q: Show that Z2 is the only group that has exactly two conjugacy classes

A: Here we know that, ℤ2=0, 1. To show that ℤ2 is the only group that has exactly two conjugacy…

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: If a group G has finitely many subgroups, does this mean that G is finite group? Justify your…

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Q: ake the minimum three groups along with subgroups (Mathematically) and verify the langrage theorem

A: Lagrangian theorem says that, If H is a subgroup of a group G then H\G. Now take, Z3={0,1,2} The…

Q: Show that the order of each element of the quotient group Q/Z is finite.

A: We have to show that the order of each element of the quotient group Q/Z is finite.

Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4

A: Given: A Sylow 2-subgroup of S4To find: Describe an isomorphism from this group to D4.

Q: (S) Is all groups of ve, glve aIl pie. about groups of order 5? (Are they always commutative).

A: Concept:

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

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Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin square…

A: G be a group of order 3 Consider G=e,a,bwhere e is the identity then the Cayley table is…

Q: A cyclic group is abelian

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Q: Show that each dihedral group is homomorphic to the group of order 2

A: To show that each dihedral group is homomorphic to the group of order 2, we need to construct a…

Q: find the number of elements in the indicated cyclic group. the cyclic subgroup of Z30 generated by…

A: To find the number of elements in the cyclic subgroup  generated by 25, you need to determine all…