Any group of order 520 is simple

Q: Prove that there is no simple group of order 525 = 3 . 52 . 7.

A: To Prove: There is no simple group of order 525=3·52·7.

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

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Q: Prove that a group of order 28 must have an element of order 2.

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Q: Prove that a group of order 63 must have an element of order 3.

A: To Prove: A group of order 63 must have an element of order 3.

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

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Q: group of order 126 cannot be simple

A: a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: Prove that a group of order 105 contains a subgroup of order 35.

A: Let G be a group of order 105. We need to show that G contains a subgroup of order 35. We know the…

Q: Prove that every group of order 330 is not simple.

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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: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

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: Prove O3 is not a group

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Q: Please help me by providing details definitions and theorems   prove that no group of order of 160…

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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: Suppose that G is a group of order pnm, where p is prime and p doesnot divide m. Show that the…

A: As per the question we are given a group G of order pnm, where p is prime and p does not divide m.…

Q: Prove that there is no simple group of order n, where 201 ≤n ≤ 235 and n is not prime.

A: We have to prove that there is no simple group of order n, where 201≤n≤235 and n is not prime. First…

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: Prove that  a group of order 1000 cannot be simple

A: Sylow's theorem: Let n be a positive integer that is not prime, and let p be a prime divisor of n.…

Q: Show that a group G of order pqr  for distinct primes, p,q and r, is  solvalbe.

A: |G| = pqr.   Without loss of generality, let p<q<r. We claim two things :  (i) There exists a…

Q: Prove that there is no simple group of order 396 = 22 .32 .11.

A: We have to prove that there is no simple group of order 396=22⋅32⋅11. We will prove this by the…

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

A: Abstract Algebra, Dihedral group

Q: Show that there is no simple group of order pqr, where p, q, and rare primes ( p, q, and r need not…

A: We know the definition of simple group, A simple group is a group whose only normal subgroups are…

Q: Prove that there is no simple group of order 120 = 23 . 3 . 5.

A: We know the definition of simple group, A simple group is a group whose only normal subgroups are…

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: Show that a group of order pn, where p is prime, is solvable.

A: Let G be a group of order p^n, where p is a prime number.   We will prove by induction on n that G…

Q: Prove that there is no simple group of order 300 = 22 . 3 . 52.

A: Introduction : A simple group is a group whose only normal subgroups are the trivial subgroup of…

Q: If a group has exactly 24 elements of order 6, how many cyclic subgroupsof order 6 does it have?

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Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…

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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: Show that (up to isomorphism) A5 is the only simple group of order60.

A: Given : G=60 To Prove : G≃A5

Q: for each of the group Z2,Z3,Z4,Z10 indicate which are cyclic. for those that are cyclic list all the…

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

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Q: True or False:  No group of order 21 is simple.

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Q: Consider a group G of order 60. Prove that G has an element of order 2. Use the fact that the order…

A: In group theory, Lagrange's theorem states that the order of a subgroup divides the order of the…

Q: Show that A5 does not contain a subgroup of order 30, 20, or 15.

A: In this question, we have to show that A5 does not contain a subgroup of order 30, 20 or 15. We know…

Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

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

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