√4. Let G be the multiplicative group of units Us = {[1], [2], [3], [4]} Zs.

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: For groups Gı and G2, determine the center of G1 × G2.

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Q: Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the group…

A: Given- G=a+b2: a,b∈ℚ and a and b are not both zero To prove- G is a subgroup of ℝ* under the group…

Q: G is the group of nth roots of unity under complex multiplication Zn is the group of integers…

A: Consider the given information.

Q: Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the…

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Q: 42. Let G be the group of 2 × 2 matrices under addition and b •{(a $): a +d=0}. d H = Prove that H…

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Q: is a group with identity (eg, eH).

A:

Q: 2. Let G = (1, 0). Decide if G is a group with respect to the operation * defined as follows: x * Y…

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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: 4) Let set G be the Cartisian product of G1 × G2, where G1 and G2 are 2 groups. Define a binary…

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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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Q: Show that the translations of Rn form a group.

A: Given: ℝn. We have to show that: The translations of ℝn form a group. We know that the orthogonal…

Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

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Q: 2. Find two distinct proper subgroups of the additive group (Z12, +}.

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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.

A: