Prove or give counter example. Every finite group of order 36 has atmost 9 subgroups of order 4 and atmost 4 subgroups of order 9

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A: From the given information, it is needed to prove or disprove that H is a subgroup of Z:

Q: ) Let G be a finite group , IGI=ps. p prime Prove that G cannot have two distinct and sep. subgroups…

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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, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

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Q: . Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

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

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Q: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…

A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…

Q: Prove that the trivial G is faithful if and only if G = {1G} representation of a nite group

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Q: Suppose that G is a finite group with the property that every nonidentityelement has prime order…

A: This can be proved by lemma that states If G is abelian with the property that every nonidentity…

Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

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

A: Given: G is a group with order 187. Theorems used: Any group of prime order is cyclic Lagrange…

Q: Let G be a cyclic group of order n. Let m < n be a positive integer. How many subgroups of order m…

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Q: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: Let G be a finite group of order p  where p is prime. Prove that G is cyclic.

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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: 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: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

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Q: Prove that the trivial representation of a nite group G is faithful if and only if G = {1G}

A: To Prove: The trivial representation of the nite group G is faithful if and only if G =1G

Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

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: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: Apply Sylow theorems to identify the correct statement Non-cyclic group of order 14 has exactly p +…

A: Please see the attachment

Q: Prove that if G is a finite group, then the index of Z(G) cannot be prime.

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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…

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

Q: Suppose that G is a group of order 105 with the property that G hasexactly one subgroup for each…

A: Given G is a group of order 105 .

Q: Apply Sylow theorems to identify the correct statement O Every group of order 65 is not cyclic…

A: Option c is the correct answer

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

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: Prove that every subgroup of nilpotent group is nilpotent

A: Consider the provided question, We know that, prove that every subgroup of nilpotent group is…

Q: 2. A Sylow 3-subgroup of a group of order 54 has order

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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: Let G be a group with |G|=187 then every proper subgroup of G is: * O Non cyclic O Non abelian O…

A: In the given question we have to choose the correct option from the given options. In the solution…

Q: Let Z denote the group of integers under addtion. Is every subgroup of Z cyclic? Why? Describe all…

A: Solution

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: Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G.  Generalize to any group of…

A: The Result to be proved is: If G is a group of order p2, where p is a prime, then either G is cyclic…

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: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

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

A: See the detailed solution below.

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

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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: 3. Let G be a finite group with an odd number of elements. Prove that G is isomorphic to a subgroup…

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Q: Show that if H and K are subgroups of an abelian group G then {hk: h element of H and k element of K…

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Q: Let G be a group of order 24. Suppose that G has precisely one subgroup of order 3, and one subgroup…

A: Theorem : If a group G is the internal direct product of subgroups H and K, then G is isomorphic to…

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.