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

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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: 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: 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: 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…