Label each of the following statements as either true or false   The order of an element of a finite group divides the order of the group.

Q: Prove that any group with prime order is cyclic.

A: Given, Any group with prime order. let o(G)=p (p is a prime number) we assure that G has no subgroup…

Q: Prove that there is no simple group of order p2q, where p and q areodd primes and q > p.

A: Let G be a group. |G| = p2q, where p and q are odd primes and q > p. The first Sylow Theorem: A…

Q: Prove that for a fixed value of n, the set Un of all nth roots of 1 forms a group with respect to…

A: Fix an n∈ℕ. Let U=u∈ℂ | un=1 .  Note that U⊂ℂ* and ℂ* is a group under multiplication. Let u,v∈U…

Q: Let G be a finite cyclic group and a be an element of G of order 8 then the order of G would be: O…

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Q: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m> 1, the…

A: Since part b,part are independent of part a,as per the guidelines I am answering the part a only.…

Q: Prove A3 is a cyclic group

A: We know that If G be a group of prime order then G is cyclic group

Q: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m > 1,…

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Q: that a and xax^-1 have same order for all x belongs G.

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Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

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Q: If G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…

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Q: Let S, be the symmetric group and let a be an element of S, defined by: 1 2 3 4 5 67 8 9 ) B = (7…

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Q: Prove that, there is no simple group of order 200.

A: Solution:-

Q: Prove that there is no simple group of order 280 = 23 .5 . 7.

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Q: Let G be a group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove

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Q: For every integer n greater than 2, prove that the group U(n2-1)is not cyclic.

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Q: Label each of the following statements as either true or false. .Let H be any subgroup of a group G.…

A: Consider the given statement "Let H be any subgroup of a group G. Then H is a left coset of H in G".…

Q: Prove that in a group, (a-1)-1 = a for all a.

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

Q: for every element a of a ((a) Prove that group 6, 2 (G) is a subset of

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Q: Let G, H be cyclic groups of order 2 and 3, respectively. Find the orders of all elements of G x H

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Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

Q: Find order of all the elements of the Group = {1, −1, ?, −?} . The binary operator ∗ is defined as ?…

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Q: Q3: (A) Prove that 1. There is no simple group of order 200.

A: Simple group of order 200

Q: Prove that in a group, (a-1)¯' = a for all a.

A: To prove that in a group (a-1 )-1=a for all a.

Q: Label each of the following statements as either true or false.   The identity element in a group G…

A: In a group, the identity element is its own inverse.

Q: Prove the statement: Let g be a group, if g is abelian then (ab)2 =  a2b2.

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Q: = Prove that, there is no simple group of order 200.

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Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

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Q: Let G = (1,-1,i,-1} Prove G is a cyclic group under the multiplication operation.

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Q: Label eachof the following statement/s as either true or false.   The symmetric group Sn on n…

A: Given: The symmetric group Sn on n elements has order n. To label: The given statement is true or…

Q: Prove or disprove, as appropriate: If G x H is a cyclic group then G and H are cyclic groups.

A: GIven two groups G and H such that GxH is cyclic. True or false: G and H themselves are cyclic

Q: Label each of the following statements as either true or false. If every nontrivial subgroup of a…

A: Whether the given statement, "If every nontrivial subgroup of a group G is a p-group, then G must be…

Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: Let G be a group with p or more distinct elements of order p for some prime p. Show that G is not…

A: Given : Let G be a group which has p  elements of order p, p is a prime no.  To show G is not cyclic…

Q: Every finite group is cyclic

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Q: Let G be a finite cyclic group and a be an element of G of order 8 then the order of G would be: 2…

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Q: Let G be a cyclic group of order 33. List the orders of all its elements and how many elements have…

A: Here, G is a cyclic group of order 33.  We have to find the orders of all its elements and the…

Q: Label each of the following statements as either true or false.   Every subgroup of a p-group is a…

A: Whether the statement, "Every subgroup of a p-group is p-group" is        true…

Q: a. Every finite group is cyclic. b.Every finite group of order 36 has at most 9 subgroups of order…

A: (a) " Every finite group is cyclic"→ It is False statementCounter Example:Klein's four…

Q: This is abstract algebra:   Prove that if "a" is the only elemnt of order 2 in a group, then "a"…

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Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: Label the following statement as either true or false. Every finite group G of order n is isomorphic…

A: True Cayley's Theorem states that every group G is isomorphic to a subgroup of the group T(S) of all…

Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10 O…

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Q: Every element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.

Q: Label each of the following statements as either true or false.   The Cayley table for a group will…

A: We know that Cayley table for a group will always be symmetric with respect to the diagonal from…

Q: 2. Let a and b be elements of a finite group G. Prove that: a. a and a¯1 have the same order. b. a…

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Q: Prove that in any group, an element and its inverse have the same order.

A: Proof:Let x be a element in a group and x−1 be its inverse.Assume o(x) = m and o(x−1) = n.It is…