The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of order 6. Which ones are isomorphic? If two of the groups are isomorphic, give the relabeling explicitly. If two of the groups are not isomorphic, then give a reason.

Q: (b) Is Z7 Z10 a cyclic group? Explain. (c) Find the order of the element (2, 2) in Z7 Z10.

A: Cyclic groups and order of an element

Q: 4. The group of symmetries of the square is called Dg, and consists of elements Po: P1 P2: P3: H1,…

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Q: The icosahedral group I is the group of 60 rotational symmetries of a dodecahedron. A diagonal of…

A: Let G be a group and X a set. Then G acts on X if there exists a function ∘ : G × X → X such that…

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 G=A4. List the elements of A4 in cycle notation, and determin the order of A4.

A: Consider the group G=A4.  We have to list the elements of A4 in cycle notation, and we have to…

Q: Each of figures A and B below is obtained by adding a straight line to a regular hexagon. In figure…

A: (a) To determine if a and r are conjugate in S(A), we need to check if there exists an element b in…

Q: Show that (Q*,-) is not cyclic..

A: We use the basic definitions of cyclic groups to solve this problem. The detailed solution is…

Q: 1.e. Consider the dihedral group (D3, ◦), ie. the symmetry group of an equilateral triangle.…

A: Symmetry group of an equilateral triangle have subgroup of order 2 and it is isomorphic to Z2

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

A: Let G be a group .

Q: 4. Consider the group A4, a group with twelve elements. Note that the positive divisors of 12 are 2,…

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Q: Let Rn be the group of rotations of a regular pentagon, and let r be counterclockwise rotation by…

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Q: Find all homomorphic images of dihedral group, D3.

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Q: The point group of mer-[MA3B3] is C 2v, please lists it's symmetry elements and contain the numbers…

A: The objective of the question is to identify the symmetry elements of the point group C2v, which is…

Q: ow many subgroups of order 3 does D3 have? Write a number of subgroups only.

A: D3={1 , r , r2 , f , rf , r2f }  Or=3 Of=2    rmf=frn-m   n=3 , 0≤m≤3 O(rif)=2   0≤i≤2 3-order…

Q: (a) Consider (3,6) and (2,4) as elements of Z7xZ7. Compute (3,6) - (2,4). (b) Consider (3,6) and…

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Q: Explain why the group (Z4, +) is NOT isomorphic to the group (LI, ∘) Group (LI ∘): The symmetries…

A: If two groups are isomorphic then they should have same number of elements

Q: Decide if each of the following sets H is a subgroup or not of the group G = R XR, where the group…

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Q: 4. Which of the groups U(14), Z6, S3 are isomorphic?

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Q: 6.2 Which of the following groups are cyclic? a) Z12×Z, b) Z10x Zgs c) Z4xZ2sx Z, d) Z22XZ21 xZ6s

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