G be a group and let a e G have order 30. List the powers of a that have order 2, the rers that have order 3, and the powers that have order 5.

Q: Label the following statement as either true or false.   The alternating group A4 on 4 elements is…

A: We have to state whether the given statement is true or false : The given statement is : The…

Q: for each of the group Z2,Z4,Z6 INDICATE WHICH CYCLIC. FOR those cyclic list  all the generators?

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Q: 3. a) List all elements of (Z45.) that are of order 15. State the results used. b) Let G = be cyclic…

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Q: Let a be an element of order 8 in a group G. 6 Then order of a Select one: O a. 6 о Б.48 C. 2 d. 4

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Q: Let G be a group. Suppose G=Z6xZ4. State all subgroups of G. Categorized(classify) the…

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Q: Give examples of four groups of order 12, no two of which are isomorphic.Give reasons why no two are…

A: We have to give 4 examples of order 12, such that all are non-isomorphic to each other.

Q: Find the order of each of the following elements in the multiplicative group of units Up. a. [2] for…

A: Consider the multiplicative group U13. Find the order of element 2.   The identity of a…

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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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: f p,q and r are three distinct primes and G is a   group of order pqr, then G is not simple.

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin…

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Q: Consider the group G=A4. List the elements of A4 in cycle notation, and determin the order of A4.

A: Consider the group G=A4.  We have to list the elements of A4 in cycle notation, and we have to…

Q: Let G be a group of order n. For what value of n is the group G NOT cyclic?

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Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

Q: Suppose that a group of order 8 has exactly five elements of order 2.Identify the group.

A: Abstract Algebra, Dihedral group

Q: Let a be a group element such that |a| = 48. For each part, find adivisor k of 48 such thata.…

A: Given: a be a group element such that a=48.We need to find divisors k of 48 such that the subgroups…

Q: 10. Let G be a finite group and a € G. Use Lagrange's Theorem to prove that the order of a divides…

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Q: Suppose that a group contains elements of orders 1 through 10.What is the minimum possible order of…

A: Given that, a group contains elements of orders 1 through 10.We have to find the minimum possible…

Q: If pq and r are three distinct primes and G is a group of order par, then G is not simple.

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

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: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

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Q: Let G={aeR: a>0}. Select all of the operations from the following list such that (G, ◊) is a group.…

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Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A: Given set is G = <a, b | a^6 = b^3 = e, b^-1ab = a^3>.

Q: Let G (a) be a cyclic group of order 35. Which of the following element has order 5 Select one: a. a…

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Q: Let G = <a, b = a3 =e, b2 =e, aba-1b-1=e>. To what familiar group is G isomorphic?

A: As per the question we are given a group G such that :G = {a, b, e}, where : a3 = e, b2 = e, aba-1 =…

Q: 4*. Let a and b be elements of orders 5 and 8 in a group of order 120. List the possible orders of…

A: To list the possible orders of the subgroup generated by a and b under the given conditions

Q: How many elements of a cyclic group with order 14 have order 7?

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Q: Give a list of all groups of order 8 and show why they are not isomorphic. for this you can show…

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Q: Determine which of the following are groups under the stated operations. For those which are groups,…

A: i) the pair here we draw composition table :246824826486426246886284from above table ,we concluded…

Q: Let G be a group of order 100. If G (a) The generators of Gare: a.a.a a to 30 45 60 75 90 15 35 45…

A: Theorem Used: Let G=<g> be a cyclic group where O(G)=n. Then G=<gk > if and only if gcd…

Q: et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an…

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Q: Let G be a group of order 3 and identity e. If the other elements are a and b, use the Latin square…

A: G be a group of order 3 Consider G=e,a,bwhere e is the identity then the Cayley table is…