10) The lower class limit for the group between 11 and 19 is a) 10.5

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

A:

Q: 2. Are the groups (R, +) and (R†,') isomorphic? Justify your answer.

A:

Q: 7. Show that 4 is a subgroup of S,

A:

Q: Prove that a group of order 7is cyclic.

A: Solution:-

Q: Prove O3 is not a group

A:

Q: Let G be group, aeG ond lal=n. Show that laml = (m,n) %3D

A:

Q: Are groups Z×10and Z×12 isomorphic

A: Concept:

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

A:

Q: Show that the center of a group of order 60 cannot have order 4.

A:

Q: (c) Prove that the intersection of any three subgroups is a subgroup while the union of two…

A:

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: Prove that, there is no simple group of order 200.

A: Solution:-

Q: 3. Define Lie group.

A:

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

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

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

A:

Q: Compute the center of generalized linear group for n=4

A: To find - Compute the center of generalized linear group for n=4

Q: 8. Use Caley's table to prove that the set of all permutations on the set X = {1,2,3} is indeed a…

A: A set R together with the binary operations addition is said to be group if it satisfies the given…

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: Construct a subgroup lattice for the group Z/48Z.

A:

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

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: Let p be a prime number and (G, *) a finite group IGI= p?. How can you prove that the group (G, *)…

A:

Q: = Prove that, there is no simple group of order 200.

A:

Q: Prove that there is no simple group of order 210 = 2 . 3 . 5 . 7.

A:

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: 7. Prove that if G is a group of order 1045 and H€ Syl19 (G), K € Syl₁1 (G), then KG a and HC Z(G).…

A: As per policy, we are solving only the first Question, Please post multiple Questions separately.

Q: Show that any group of order less than 60 is cyclic

A: This result is not correct. There is a group of order less than 60 which is not cyclic.

Q: Prove that An even permutation is group w.y.t compostin Compostin function.

A:

Q: Let Ø:Z50→Z15 be a group homomorphism with Ø(x)=7x. Then, Ker(Ø)= * O {0, 10, 20, 30, 40} None of…

A:

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

A: The Oder is 24

Q: 5. (a) Ifp is a prime then the group U, has (d) elements of order d for each d dividing p- 1. (b)…

A:

Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

Q: (i). There is a simple group of order 2021.

A:

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: Show that group Un (n th unit root) and group Zn are isomorphic.

A: There are n elements in the group (Zn,+). There are n elements in the group (Un,×). There are (n!/2)…

Q: Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G…

A:

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

A:

Q: Exercise 6(@) Prove thal the add.tive Z oud Q are not isomonpkic (6) Show that there are non groups…

A: If G and H are isomorphic then G is cyclic if and only if H is cyclic. So if G is cyclic but H is…

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: 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: 8. Prove that Zp has no nontrivial subgroups if p is prime. [#26, 4.5]

A: Follow the steps.

Q: Show that * defined on Z by a*b=|a+b| is not a group. (Hint: identify and show the group property…

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: 6.7 Construct a nonabelian group of order 16, and one of order 24.

A: