Suppose that G is a group of order 105 with the property that G hasexactly one subgroup for each divisor of 105. Prove that G 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: Let G be a cyclic group with more than two elements: 1. Prove that G has at least two different…

A: Given: Let G be a cyclic group.

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: abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp×Zp

A:

Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let G be a finite cyclic group of order 20, and a in G. Then one of the following is possible order…

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Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

A: We know that a finite group G is said to be cyclic if and only if  there exist an element in G such…

Q: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…

A: Q1.  Third option is correct. Q2. Second option is correct.

Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * True False

A: let H be a cyclic subgroup of a group G.

Q: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

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

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Q: Prove that the intersection of two subgroups of a group G is a subgroup of G.

A: We will prove the statement.

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

A:

Q: Suppose that G is a finite Abelian group. Prove that G has order pn,where p is prime, if and only if…

A: We have given G is a finite Abelian group, then we have to prove that G has order pn,where p is…

Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Find a proper subgroup of the group of integers ? under addition and prove that this subgroup is…

A: Solution: The objective is to find a proper subgroup of the group of integers and to show that this…

Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of the order…

A: Here given that G is a finite Abelian group that has exactly one subgroup for each divisor of the…

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: Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

Q: Let G be a group and suppose that G only has finitely many distinct subgroups. Show that |G| < ∞.

A: If possible  let G is of infinite  order .In particular let  G = {a1,a2,...,an,...} .Let us consider…

Q: Suppose that G is a group with more than one element and G has noproper, nontrivial subgroups. Prove…

A: G is a group with more than one element and G has noproper, nontrivial subgroups. We need to prove…

Q: 5. Suppose G is a group of order 8. Prove that G must have a subgroup of order 2.

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Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

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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: Let G be a group with |G|=187 then every proper subgroup of G is: * Non cyclic None of these Сyclic…

A: Given that G is a group and G=187 i.e., number of elements in G is 187.  First we have to find the…

Q: If N is a subgroup of an Abelian group, prove that is Abelian. N |

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Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

Q: Prove that any two Sylow p-subgroups of a group G are conjugate.

A: To prove that any two sylowp-subgroup of a group G are conjugate

Q: Suppose that G is a finite Abelian group that has exactly one subgroup for eah divisor of |G|.  Show…

A:

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

A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…

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

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Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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Q: Prove that every group of order 675 has a normal Sylow 5-subgroup.

A: Theorem: If number of Sylow p-subgroup of a group G is unique, then that subgroup is normal. Let G…

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

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Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…

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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: If G is a cyclic group, prove for subgroup N that G is a cyclic N

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

A:

Q: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

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Q: Suppose that G is a finite Abelian group that has exactly one subgroup for each divisor of |G|. Show…

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Q: Suppose G is a group in which all nonidentity elements have order 2. Prove that G is abelian.

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Q: Let a and b be nonidentity elements of different orders in a group Gof order 155. Prove that the…

A: Given that Let a and b be nonidentity elements of different orders in a group G of order 155. Note…

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: Suppose G = (a) is a cyclic group of order 6. Find all the subgroups of G and list the elements in…

A: We have to find all the subgroups of G and list the elements in each of these subgroups.