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

A: This is a problem of Group Theory.

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

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A: To find- Write the following permutations in cycle notation. 1234542513 1234514325

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…

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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: 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 .

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Q: the indicial equation of the differential equation xy'' -x2y'+(1+2x)y=0

A: Question from differential equations.

Q: The inverse of each element in the group ({1, 3, 5, 7}, ∙8)

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A: Use the formula of o(a^r) , if o(a) is given.

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

Q: 4

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

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A: We have to prove that given statement:

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A: As  -1 &lt; x And  2 &lt; y Then   -1+2 &lt; x+y I. e. 1 &lt; x+y

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: 34. Suppose a and b are group elements such that |a| = 2, b ‡ e, and aba = b². Determine [b].

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Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________

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

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A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

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