1. Let G.*) be a group and a EG Suppose that a*a = a Prove or disprove that a must be the identity element.

Q: An is the alternating group on n letters. What is |An|?

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Q: group , Poq EG, and p"qp=q? + q! pq=p's prove that Let G be a

A: I have done algebraic calculations.

Q: Let a be a group element and lal = o0. Complete the following statement: la'l = la'l if and only if…

A: Given that a be a group element and a=∞. We need to complete the statement: ai=aj if and only if…

Q: 19. Prove that the group (Z/11Z)* is cyclic, and find a generator.

A: We have to prove that the group (Z/11Z)* is cyclic, and find a generator.

Q: Prove or disproe: If C is a groun and a hsC then (ab)-1 h-1a-1

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Q: 30. Prove the right and left cancellation laws for a group G; that is, show that in the group G, ba…

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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: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

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Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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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: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

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: 3. Prove or disprove: If G is a group, (g¬')-' = g.

A: Consider the given information: Let G is a group. To show that (g-1)-1=g

Q: Let G=.  How many elements does G have?  To what familiar group is G isomorphic?

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Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…