Prove that a group of order 15 is cyclic

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 a group of order 375 has a subgroup of order 15.

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

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Q: Prove that every group of order 1225 has a normal abelian Sylow 5-subgroup.

A: Since not a particular question asked as per guidelines solution to only first question  is given…

Q: Prove that a group of order 7is cyclic.

A: Solution:-

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 that a group of order 175 is Abelian.

A: Let G be a group of order 175 We first try to rewrite 175 as prime factorization as follows: 175 =…

Q: All groups of order three are isomorphic.

A: All groups of order three are isomorphic.

Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

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: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

Q: Let G be a group of order 24. Show that G is solvable

A: Let G be a group of order 24. Show that G is solvable

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

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Q: Let G be a group of order 60. Show that G has exactly four elementsof order 5 or exactly 24 elements…

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Q: Prove that a group of order 12 must have an element of order 2.

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Q: Prove that any two groups of order 3 are isomorphic.

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Q: Every cyclic group is a non-abelian group.

A: False, all cyclic groups are abelian group. This is true

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: prove that Every group oforder 4

A: Give statement is Every group of order 4 is cyclic.

Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: Prove that every subgroup of an abelian group is abelian

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Q: Show that any group of order 4 or less is abelian

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

A: The prime factors of 525 are 3, 5 and 7. So there are proper normal subgroups of order either 3,5 or…

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: 10. Prove that any cyclic group is abelian.

A: As you are asked multiple questions but as per our guideline we can solve only one. Please repost…

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: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

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

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Q: Prove that a factor group of a cyclic group is cyclic

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Q: Show that any group of order less than 60 is cyclic

A: This result is not correct. There is a group of order less than 60 which is not cyclic.

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

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Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a…

A: Consider the x is the generator of cyclic group H for xn∈H, ∅(x)=y As a result, For all members of…

Q: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.

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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: Any group of order 520 is simple

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

Q: Prove that any group of order 75 can have at most one subgroup of order 25.

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Q: Prove that every abelian group is cyclic.

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Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…

Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

Q: Let M be a group (not necesarily an Abelian group) of order 387. Prove that M must have an element…

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Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

Q: Prove that every group of order 78 has a normal subgroup of order 39.

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Q: Prove that every group of order (5)(7)(47) is abelian and cyclic.

A: See the attachment

Q: Prove that every group of order 375 has a subgroup of order 15.

A: According to the given information it is required to prove that every group of order 375 has a…

Q: Show that the translations of Rn form a group.

A: we have to show that   the translations of Rn form a group   We know that