8. a) Find all possible isomorphism classes for abelian groups of order 108. b) Find all possible isomorphism classes for abelian groups of order 1176.

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

A:

Q: Let m, n ɛ Z+ such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: Given that m,n∈ℤ+ such that gcdm,n=1. Suppose that there are M and N abelian groups, up to…

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: 4. Give an example of a pair of dihedral groups that have the same number of conjugacy classes as…

A: We know that Dihedral group is denoted by Dn. Its order is 2n. Here, to find the pairs of Dihedral…

Q: 9. Prove that a group of order 3 must be cyclic.

A:

Q: Give a careful proof using the definition of isomorphism that if G and G` are both groups with G…

A: Definition :Let, G and G' be any two group . Then a map φ: G →G' is said to be a homomorphism if φ(a…

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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: 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: 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…

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Q: 3. Find all Abelian groups (up to isomorphism) of order 100. From these isomorphism classes…

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

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: Use the theory of finite abelian groups to find all abelian groups of order 4. Explain why there are…

A: we know that if G is finite group whose order is power of a prime p thenZ(G) has more than one…

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…

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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: 5: (A) Prove that, every group of prime order is cyclic.

A:

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.

A:

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

A:

Q: Which abelian somorphic to groups subyraups of Sc. Explin. are

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

A:

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

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A: Find Aut(Z20) by using the fundamental theorem of Abelian groups

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: List six examples of non-Abelian groups of order 24.

A: The Oder is 24

Q: Give an example of two non-homomorphic finite abelian groups that have the same number of elements…

A: We need to find an example of two non holomorphic finite abelian groups which have the same number…

Q: Characterize those integers n such that the only Abelian groups oforder n are cyclic.

A: According to the question,

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: Show that there are two Abelian groups of order 108 that haveexactly 13 subgroups of order 3.

A: Aim: There are two Abelian groups of order 108 that have exactly 13 subgroups of order 3.

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

A:

Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

Q: Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an…

A:

Q: List all abelian groups (up to isomorphism) of order 600

A: To List all abelian groups (up to isomorphism) of order 600

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: 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: Find the number of isomorphism classes of the abelian groups with order 625. Yanıt:

A: We have, 625 = 5⁴    Note : For any prime p, there are as many groups of order pk as there are…

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: (5) Make a list of all group homomorphisms from Z4 → Z8.

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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: Find all possible isomorphism classes for abelian groups of order 108.

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.