1. Let G be a group. For any g E G and n = Z, we define if n > 0; g" = 99, n g • 9 |n| 9 if n = 0; if n < 0. Assuming the exponent laws for positive integer exponents, prove the following exponent laws for any integer exponents. (a) gngm = gn+m for all g € G and all n, m € Z. (b) (gn) m = gnm for all g G and all n, m € Z. (c) If g, h E G and gh = hg, then (gh)n = gnh" for all n € Z.

Q: 2. Give a multiplication table for the group U(12).

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Q: Let G:- [0, 1) be the set of real numbers x with 0<x<1. Define an operation + on G by ** y={x+y if…

A: Solution:-  Given   G=0,1   be the set of real numbers xwith  0≤x&lt;1.  The operator ∗on G is given…

Q: Let (a1a2a3a4) and (a5a6) be disjoint cycles in S10. Show that there isno element x in S10 such that…

A: Given: a1a2a3a4 and a5a6 are disjoint cycles in S10.We need to show that there is no element x in…

Q: 1. Is * commutative? or Is * associative? 2. Compute a*(b*c). 3. Compute (a*b)*c.

A: This is a problem of Group Theory.

Q: 2. Let G be a group. Recall we showed in class that if |G| is finite, then o(g) is finite for all g…

A: be a group.If  is finite then  is also finite where We have to show the converse of the statement…

Q: If & be the additive group. of real numbers and Rt the multiplictive group of positive real numbers…

A: Given that  be the additive group of real numbers and  be the multiplicative group of positive real…

Q: R\{-1} by a * b = a + b+ ab. Show that (R \ {-1},*) is an 3. Define the operation abelian group. *…

A: The solution for the asked part , is given as

Q: 2. Let a € R and n € Z0. Show that of the numbers a, 2a, distance from an integer is at most 1/n.…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Let G be the set of nonnegative integers. Is a group, |a - b| for all a, b, EG? where a * b =

A: We will check whether it is group or not.

Q: Suppose S represents any group that has the order |S| = R = 1,234,567, and suppose there is an n…

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Q: 1. Build composition table for a dihedral group D4 (Hint: You might need read additional materials…

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Q: Suppose that X and Y are subgroups of G if |X|-32 and IYI=48, then what is the best possible of Xn…

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Q: Let G be a group and let H< G. If [G: H] = 16 and |H| = 21, then what is |G|?

A: The expression, G:H can be written as GH .

Q: 1. Consider the following groups G and H, K < G. If G is isomorphic to H × K, give an isomorphism v…

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Q: Exercise 4.2. Let G be a group and a, b = G. Suppose that ab has finite order n. Prove or disprove…

A: Step 1: Step 2: Step 3: Step 4:

Q: 3. Infinite, abelian (and additive), and so that for every element a, a+a=0. (Hint: Think of…

A: Note the question asks us for an example of an infinite group such that sum of two elements is…

Q: Let G be a group and a E G. Define C(a) = {x € G|ax = xa, for all a E G}. Prove that C(a) < G.

A: A nonempty subset H of a group G is said to be a subgroup of G, if it satisfies the following…

Q: 4.1.7. Let G = Dg act on N = D8 by conjugation. Make an 8 × 8 table, where the rows are indexed by…

A: It is given that G=D8 act on Ω=D8 by conjugate. So it follows that, D8=e, r, r2, r3, s,sr, sr2,…

Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

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Q: What is the formula used to calculate degrees of freedom for a t-test for dependent groups? Group of…

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Q: 0^0, (zero to exponent zero) is ____ Group of answer choices a. infinity b. 1 c. zero d.…

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Q: 2. Let (G. .) be a group such that a.a = e for all a EG. Show that G is an abelian group.

A: Definition of abelian group : Suppose &lt;G, .&gt; is a group then G is an abelian if and only if…

Q: 6. Let f and g be two arithmetic functions with g(n) = Σan f(d). If g is multiplicative, then so is…

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Q: 3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b.…

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Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: 4

A: We use theorem of order of group and order of elements of group .

Q: . Let G be a group, let g e G, and let H - G. Suppose that the element Hg E G/H has order n. Show…

A: We have to prove that given statement:

Q: Question 1. Let G = Q be the set of all rational numbers, and * = + be the usual addition operation…

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Q: Let G be a defimed by is a group action. group and A= G. Show a*X = ANA' , aXE G that *

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Q: (a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove…

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Q: 1)Let G be a group and H be a subgroup of G then, for any a, b e G,a = b mod H + (a + b-1) H = H.…

A: Given: To show the correct option is true or false as follows,

Q: 4. Let G = {a+b√2, a, b E Z+}, * is ordinary multiplication. Is G a group?

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Q: A student takes a true-false test that has 12 questions and guesses randomly at each answer. Let X…

A: In this case, we can use the binomial distribution  Therefore, the binomial distribution is given…

Q: 1. Define the set G = {x E Zž1 | x³ = 1 (mod 31)}. (a) Prove that G is a group. (b) Find an element…

A: Given - Define the set G = x ∈ ℤ31* | x5 ≡ 1 mod 31. To find - Prove that G is a group. Find an…

Q: 0.1.9. Find a group G, with subgroups H and K, such that H◄K, K◄G, but H not normal in G.

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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…