What is the smallest positive integer n such that there are exactlyfour nonisomorphic Abelian groups of order n? Name the fourgroups.

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

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Q: ii) Does there exist a group G such that G/[G,G] is non-abelian? Give an example, or prove

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Q: 6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and…

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Q: Q3\ Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

A: (G, *) be a finite group of prime order   To prove (G, *) is an abelian group

Q: Suppose that G is an Abelian group of order 35 and every element of G satisfies the equation x35 =…

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Q: No. of isomorphic subgroup of group of integers under addition is: -

A: As we know group of integers under addition is (Z,+)

Q: Show that the group of permutations Σ2 is abelian. Then show that Σ3 is not. Writing up the group…

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Q: G is abelien group Shu subset { EGX=e} is a Wth ideotly e. wanna Subgraup of G.

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Q: Give an example, with justification, of an abelian group of rank 7 and with torsion group being…

A: consider the equation

Q: The symmetry group of a nonsquare rectangle is an Abelian groupof order 4. Is it isomorphic to Z4 or…

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Q: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

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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: b. Find all abelian groups, up to isomorphism, of order 360.

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Q: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

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Q: How many subgroups of a not abelian group of order 6 is non-cyclic? Select one:

A: Given: The order of the group = 6.

Q: (1) State the structure theorem for finite generated Abelian group. Using this theorem to classify…

A: To find the all finite generated Abelian group of order 120: Structure theorem for finite generated…

Q: 4. List all of the abelian groups of order 24 (up to isomorphism).

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Q: Please help me understand the following question and please explain the steps. Picture is below

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Q: Show that the units in Commutative ring unit element forms with an abelian group wv.t multipli…

A: let the two units of ring  R are u  and v let u-1 and v-1 also belong to R. therefore, uu-1=1 and…

Q: b. Find all abelian groups, up to isomorphism, of order 720.

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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 ca say that G is:…

A: an abelian group, also called a commutative group, is a group in which the result of applying the…

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: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

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Q: Suppose that G is an Abelian group with an odd number of elements.Show that the product of all of…

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Q: (1) Z/12Z

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Q: Without using the structure theorem for finite abelian groups, prove that a finite abelian group has…

A: Given: Let G be a finite abelian group. To prove: G has an element of order p, for every prime…

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

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: 8. Prove that Q is not a subcartesian product of infinite cyclic groups.

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Q: Find all possible isomorphism classes for abelian groups of order 1176.

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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: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

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

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Q: ) Describe, up to isomorphism, all abelian groups of order 300 = 22 . 3 - 52 using the Fundamental…

A: Solution: Given : Group of order 300

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: What is the smallest positive integer n such that there are two nonisomorphicgroups of order n? Name…

A: Non-isomorphic groups: Groups that have different Sylow-2 groups are non-isomorphic groups.

Q: Given two examples of finite abelian groups

A: Require examples of finite abelian groups.

Q: Let (G,*) be a finite group of prime order then (G,*) is a cyclic

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Q: Show that there are two Abelian groups of order 108 that haveexactly one subgroup of order 3.

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Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6.  ℤn denotes the abelian cyclic group of order n. Justify…

A: To show : ℤ12 is not isomorphic to ℤ2⊕ℤ6 Pre-requisite : P1. A group G is said to be cyclic if there…

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: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

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: Every abelian cy elic O True O False group is

A: to check  whether every abelian group is cyclic or not? proof let a euler group U8=1,3,5,7 let…

Q: 10. Prove that all finite groups of order two are isomorphic.

A: Here we use basic definitions of Group Theory .

Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

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