Consider the group G = ℤ9 ⨉ ℤ27 where the operation is done componentwise, that is (a, b) * (c, d) = (a +9 c, b +27 d), for all (a, b), (c, d) in G. (a) Find the elements of the subgroup <(1, 3)>. (b) Compute [G : <(1, 3)>]. (c) Find the elements of the left coset (1, 6) + <(1, 3)>.

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

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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: -) Write the Cayley table for the group U(15).

A: U(15) is the set of all elements in ℤ15={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} ; x≡x ,such that…

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

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Q: Consider the groups U(8) and Z4: (1) Determine the identity element in the group U(8) ×Za: (ii)…

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Q: List all the subgroups of Z, including the “obvious" or “trivial" ones. Explain how you generated…

A: Group: A group is a set G together with a binary operation a, b↦a∗b:G×G→G satisfying the following…

Q: To which group is Z12/ ker f isomorphic to? (a) Z₁ (b) V4 (c) D4 (d) A₁

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Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

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)/&lt;9&gt; Concept: Order of factor group…

Q: the group (a) find all possible orders of elements in G; (b) for each possible order of an element…

A: (ℤ/364ℤ,+) is isomorphic to (ℤ364,+). All possible orders of (ℤ364,+) are all possible divisors of…

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

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Q: In the group U (16) compute, (a) 5-7, and (b) 3-1

A: M(16) refers to th e multiplication group of non-negative integers (mod 16) that are less than 16…

Q: 5. Let S : R³ R2 and T: R² ->> X1 2 X1 $8-0-0-- .T = X2X3 S x2 R2 be defined by X3 X1 X2 = x1 + 3x₂…

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Q: 5. Consider the "clock arithmetic" group (Z15,0) a) Using Lagrange's Theorem, state all possible…

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Q: Consider the group A₁. Let H = {e, (12)(34), (13)(24), (14)(23)}. (a) Show that H is a subgroup of…

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Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | |   2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…

Q: b. Find the center and the commutator subgroup of S2 × Z7.

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