6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and prove it).

Q: In (Z12, +12) , H, = {0,3,6,9} and H2 = {0,3} are tow subgroups of the group (Z12, +12), but (H, U…

A: We have given   H1=0,3,6,9 and H2=0,3 are two subgroups of the group Z12, +12. We can see here…

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: (d) Show that Theorem 1 does not hold for n = 1 and n = 2. That is, show that the multiplicative…

A:

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

A:

Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

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: Prove O3 is not a group

A:

Q: Are groups Z×10and Z×12 isomorphic

A: Concept:

Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

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

A:

Q: 4.14. Show that an element of the factor group R/Z has finite order if and only if it is in Q/Z.

A: Any rational number can be written in the form p/q where p and q are relatively prime integers.Since…

Q: Show that if n > 6, then the symmetric group Sn contains an element of order greater than n

A: According to the problem, we have

Q: 25. o: Z4 → Z12

A: Homomorphism : Let us consider a map f: V→W then f is said to be homomorphism if for all v,u∈V…

Q: How many nonisomorphic abelian groups of order 80000 are there?

A:

Q: Show that if a group has an even number of elements then there is a element A other than unity such…

A: We have to prove that if a group has an even number of elements then there is aelement A other than…

Q: How many ways can a group of five be chosen to work on a project? As in Example 9.5 4, since the of…

A: Given that, the computer programming team has 9 members.

Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

A:

Q: What is the order of the element $(\overline{2}, \overline{9})$ in $Z_{4} \times U_{10}$ is (…

A: The set ℤ4 ,i.e the set of congruence modulo 4 is an additive group. Add the element 2 with itself…

Q: 1. Determine all subgroups of the group (U13, ·)

A: The sub group of U13 is to be determined.

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

A:

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: The Sylow theorems are significant in the categorization of finite simple groups and are a key…

Q: How many elements of a cyclic group with order 14 have order 7?

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

A:

Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

A:

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

A:

Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: Show that there are no simple groups of order 255 = (3)(5)(17).

A:

Q: 3. List all elements of the cyclic subgroup of Z12 generated by 5

A: Solving

Q: (S) Is all groups of ve, glve aIl pie. about groups of order 5? (Are they always commutative).

A: Concept:

Q: determine whether the binary operation * defined by a*b=ab gives a group structures on Z. if it is…

A:

Q: Let φ : Z2 → Z9 be defined by φ(n) = n (mod 2). Is φ a group homomorphism?

A: Let φ:ℤ2→ℤ9 defined by φ(n)=n(mod 2) We have ℤ2=0¯,1¯, ℤ9=0¯,1¯,...,8¯

Q: Theorem: Any non-commutative group has at least six elements

A:

Q: 27. Prove or disprove that each of the following groups with addition as defined in Exer- cises 52…

A: Let G = Z2xZ4  i.e G = { (0,0),(0,1),(0,2)(,0,3),(1,0),(1,1),(1,2),(1,3)}  Order of G = 8

Q: for each of the group Z2,Z3,Z4,Z10 indicate which are cyclic. for those that are cyclic list all the…

A:

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

A: The Oder is 24

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: 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: 24, Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then a and b are…

A: Given:  Let G be a group. ZG be its center.  We know that ZG=z∈G: ∀g∈G,zg=gz ....i First we will…

Q: Prove that there are exactly five groups with eight elements, up to isomorphism.

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: 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: determine whether the binary  operation * defined by a*b=ab gives group structure of Z. if it is not…

A:

Q: 2) Determine whether or not the groups Z10 × Z4 and Z, × Z20 are isomorphic. Justify your answer.

A:

Q: True or false? Every group of 125 elements has at least 5 elements that commute with every element…

A: Let G be a group whose order is 125 ⇒G=125=53 Center of a group G ( ZG ) is the set of all those…

Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…

A:

Q: 8. Show that (Z,,×s) is a monoid. Is (Z.,×6) an abelian group? Justify your answer

A: Note: since you have posted multiple questions . As per our guidelines we are supposed to solve one…

Q: 2) Given example of an infinite group in which every nontrivial subgroup is infinite.

A: Let G=a be an infinite cyclic group generated by a, whose identity element is e. Let g∈G, g≠e,…

Q: 1,) and (G2,*) be two groups and →G2 be an isomorphism. Then *

A: given that G1,. and G2,*are two groups  and φ:G1→G2  be an isomorphism