b. Find all abelian groups, up to isomorphism, of order 720.
Q: Give all the possible elementary divisors of a group of order 40.
A: 40=23×5 So, the possible elementary divisors of the group are 2,2,2,5,2,4,5 and 8,5
Q: Prove that every group of order 330 is not simple.
A:
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: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D
A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…
Q: All groups of order three are isomorphic.
A: All groups of order three are isomorphic.
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: Prove A3 is a cyclic group
A: We know that If G be a group of prime order then G is cyclic group
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Q: For each of the following values of n, describe all the abelian groups of order n, up to…
A: Given information n=10.
Q: For each of the following values of n, describe all the abelian groups of order n, up to…
A: For positive integer n, let Cn denote a cyclic group of order n. If G is an abelian group of order…
Q: Give an example of a p-group of order 9.
A: Given, Give an example of a p-group of order 9.
Q: Consider the group 6 * (x ER such that x0) under the binary operation identity element of G is e =…
A: Thanks for the question :)And your upvote will be really appreciable ;)
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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: How many nonisomorphic abelian groups of order 80000 are there?
A:
Q: Every cyclic group is a non-abelian group.
A: False, all cyclic groups are abelian group. This is true
Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…
A:
Q: Prove that, there is no simple group of order 200.
A: Solution:-
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: 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: Q7/ Find all possible non-isomorphic groups of order 77.
A:
Q: If a is an element of order 8 of a group G,
A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e where e is an identity…
Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…
A:
Q: 5: (A) Prove that, every group of prime order is cyclic.
A:
Q: 4. Let G be a finite group of order 4 containing no element of order 4. Explain why every…
A: See below
Q: Find all possible isomorphism classes for abelian groups of order 1176.
A:
Q: 5. How many automorphisms does Klein's 4-group have?
A: No of automorphism of Klein's 4-group: K4={e,a,b,c} f1=eabceabc = I f2=eabceacb = bc f3=eabcecba =…
Q: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…
A:
Q: = Prove that, there is no simple group of order 200.
A:
Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.
A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…
Q: For each of the following values of n, describe all the abelian groups of order n, up to…
A: Need to evaluate all the abelian group of order 36.
Q: 9 Find all isamorphism classes of abelian groups of arder 72.
A: We have find isomorphism classes of abelian group of order 72:
Q: Find the number of isomorphism classes of the abelian groups with order 16.
A: The order of abelian groups = 16
Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.
A:
Q: Determine the class equation for non-Abelian groups of orders 39and 55.
A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.
Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group
A:
Q: For each of the following values of n, describe all the abelian groups of order n, up to…
A: We use the following result here. Result: Let n=pq for some prime numbers p and q with p<q. If p…
Q: List all abelian groups (up to isomorphism) of order 600
A: To List all abelian groups (up to isomorphism) of order 600
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: What is the smallest positive integer n such that there are exactlyfour nonisomorphic Abelian groups…
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Q: 300Can someone please help me understand the following problem. I need to know how to start the…
A: G is the abelion group of order 16. It is isomorphic to,
Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…
A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…
Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…
A: An automorphism of a group is the permutation of the group which preserves the property ϕgh=ϕgϕh…
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: suppose H is cyclic group. The order of H is prime. Prove that the group of automorphism of H is…
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Q: 10. Prove that all finite groups of order two are isomorphic.
A: Here we use basic definitions of Group Theory .
Q: Consider the alternating group A4. Identify the groups N and A4 /N up to an isomorphism.
A: Consider the alternating group A4. We need to Identify the groups N and A4 /N up to an isomorphism.…
Q: Show that a group of order 77 is cyclic.
A:
Q: Give a list, up to isomorphism, without repetition, of all abelian groups of order 72.
A:
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- Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.Prove that if r and s are relatively prime positive integers, then any cyclic group of order rs is the direct sum of a cyclic group of order r and a cyclic group of order s.
- Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.