Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is |ab|?

Q: Is the set ℤ+ under addition a group? Prove your answer using the properties of the group. Note:…

A:

Q: Suppose G is a cyclic group with an element with infinite order. How many elements of G have finite…

A: Suppose G is a cyclic group with an element with infinite order.  It means that order of group is…

Q: Theorem :- Let (G,-) be a group then :- 1- (Hom(G),) is semi group with identity. 2- (A(G),) is…

A:

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

A:

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will  proof the statement otherwise disprove we taking the…

Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: Define Group theory ?

A: To define group theory

Q: What is the smallest positive integer n such that there are three nonisomorphicAbelian groups of…

A:

Q: Express U(165) as an internal direct product of proper subgroups infour different ways

A:

Q: does the set of polynomials with real coefficients of degree 5 specify a group under the addition of…

A:

Q: Explain why the only simple, cyclic groups are those of prime order.

A: Proof: Let G be a simple group with |G|>1. We want to prove that G is a cyclic group of prime…

Q: Let F denote the set of first 8 fibanacci number .Then convert F into a non abelian group by showing…

A:

Q: ab 2. Show that (Z, *) is an abelian (commutative) group, where' is defined as a*b = and Z is the…

A:

Q: Find an isomorphism from the group G = to the multiplicative group {1, i, – 1, – i} in Example 3 of…

A:

Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

A:

Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

A:

Q: Every group of order 4 is cyclic. True or False then why

A: Solution

Q: 3. Define Lie group.

A:

Q: Determine all cyclic groups that have exactly two generators.

A:

Q: 6. Embed the group Qs into the SU(2).

A: Given: Q0=e,i,j,k  e-2=e, i2=j2=k2=ijk=e, Where, e is the identity element and e commutes with the…

Q: 1) (Z,, +,) is a group, [3]- is 2) 11 = 5(mod----) 3) Fis bijective iff

A:

Q: chow that An s a Group with respect, to Cooplesition of functjon.

A:

Q: Is it possible to find a group operation e on a set with 0 elements? With 1 element? Explain why or…

A: The question is :: is there possible to find a group operation on a set of 0  element? Or with 1…

Q: Show that a subgroup of a solvable group is solvable.

A:

Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is a…

Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

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: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

A:

Q: Show that every group of order < 60 is solvable.

A:

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

A: Writing a permutation σ∈Sn as a product of n disjoint circles. i.e σ=τ1,τ2,τ3,…τk The order of σ is…

Q: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

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

A:

Q: ake the minimum three groups along with subgroups (Mathematically) and verify the langrage theorem

A: Lagrangian theorem says that, If H is a subgroup of a group G then H\G. Now take, Z3={0,1,2} The…

Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank&gt;1 has trivial center

Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

A:

Q: EQ6/ Give example of a group such that has element of order 7 and nurmal subgruup of Order 3.

A: A detailed solution is given below.

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

A: The Oder is 24

Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

A:

Q: Show that the groups Z8xZ20xZ12 and Z120xZ4xZ4 are isomorphic by define a one-one and onto map? what…

A: We will use the basic knowledge of groups and abstract algebra to answer this question.

Q: 8. Give an example of a group G where the set of all elements that are their own inverses does NOT…

A: Let, G,. is a group. Let, G={1,7,17,2,12,3,13} Let, H be a subgroup of G where H={1,7,17,2,12}

Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn

A:

Q: Determine the galois group

A:

Q: 2. Is the set Z3 = {0,1,2} form a group with respect to addition modulo 3 how about to…

A: We use caley's table to verify the properties of a group.

Q: Let F denote the set of first fibonacci number .Then convert into a non abelian group by showing all…

A:

Q: Check whether a group of order 156 is simple or not.

A: Definition: A group G is said to be simple if the only normal subgroups of G are the trivial group e…

Q: Let 0:Z50-Z15 be a group homomorphism with 0(x)=4x. Then, Ker(Ø)= {0, 10, 20, 30, 40)

A:

Q: Find the outer set of points for group S.

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Let F denote the set of first 8 fibonacci number.Then convert into a non.abelian group by showing…

A: