5. Let G be the multiplicative group {[2], [4], [6], [8]} = Z10-

Q: Compute the factor group (Z6 x Z4)/(([2]6, [2]4))

A:

Q: Which of the following is group under multiplication? O a. Q-{0} O b. None of them О с. (-{1} O d.…

A:

Q: Prove that an Abelian group with two elements of order 2 must have a subgroup of order 4.

A: Abelian groups are groups where elements are commutative. That is, ab=ba

Q: What is the order of the factor group Z60/<15>?

A:

Q: take the groups Z4 and Z5, what will be the elements of Z4 x Z5?

A:

Q: Explain why S8 contains subgroups isomorphic to Z15, U(16), and D8.

A:

Q: 7. [10] Let G be a group, with x € G and o (x) = 48. Give the order of the following elements of G:…

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Q: Find the regular permutation group isomorphic to the multiplicative group G={1,-1,i,-i}.

A:

Q: 3. In each part, find the order of the given element in the given group. (a) i= √-1 in the…

A:

Q: 19. Prove that the group (Z/11Z)* is cyclic, and find a generator.

A: We have to prove that the group (Z/11Z)* is cyclic, and find a generator.

Q: means a cyclic group but 3+i can't turn into 3-i with a positive integer exponent.

A: Please see the answer in next step.

Q: 12. Find all subgroups of Z2×Z4.

A:

Q: How many elements of order 5 might be contained in a group of order 20?

A: using  third Sylow Theorem

Q: Express the following as products of cyclic gr a) Z140 ») U (140) =) Z400 H) U (400).

A: To express the given groups as products of cyclic groups, we need to find the prime factorization of…

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

A:

Q: The group (U20, X20) = {(1, 3, 7, 9, 11, 13, 17, 19), ×20} has subgroups H = {1,3,7,9} and K =…

A:

Q: Find the order of the factor group U(16)/(9)where U(16)={1,3,5,7,9,11,13,15} with operation…

A: Since , we have to find order of the factor group U(16)/<9> Concept: Order of factor group…

Q: Find the order of the given factor group (Z3 x Z6)/

A: If G be a group, and H be a subgroup of G, then the order of the factor group G/H is given by…

Q: Find the index of in the group Z24

A:

Q: 2.15. (a) Prove that GL2 (Fp) is a group. (b) Show that GL2 (Fp) is a noncommutative group for every…

A: And

Q: In the group Z8 compute, (a) 6+7, and (b) 2-1.

A:

Q: Let G = Z4 x Z6. Compute the factor groups G/((2, 3)) and G/((3, 3)). (In each case, write the…

A:

Q: 11. Prove that the set of all 2 × 2 matrices with entries from R and de- terminant +1 is a group…

A: To show that the set of all 2×2 matrices with entries from R and determinant +1 is a group under…

Q: Find all the elements of a factor group Z3 X Z4 /

A: We have to find the all the elements of a factor group ℤ3×ℤ4/2,2 Let the group H=ℤ3×ℤ4 Then order of…