Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = –S, |S| = 3 and (S) = Zn. %3D (a) Prove that n is even number. (b) If n > 6, prove that Cay(G, S) is a normal Cayley graph.

Q: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

A:

Q: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

Q: Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements…

A: In a group G, two element g and h are called conjugate when h = x g x-1 for some x ∈ G For an…

Q: Let G be the subgroup of GL3(Z₂) defined by the set 100 a 10 bc1 such that a, b, c Z₂. Show that G…

A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

Q: (G, .) a group such that a.a = e for all a EG.Show that G is an abelian group. Let be

A:

Q: Let G be a cyclic group of order 135. The number of generators of G is The number of elements of…

A:

Q: A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is…

A: The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If we…

Q: (2) of order 5 is in H. Let G be a group of order 100 that has a subgroup H of order 25. Prove that…

A:

Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

A:

Q: Show that Z is not isomorphic to Zž by showing that the first group has an element of order 4 but…

A: The given groups: Z5x and Z8x That is, Z5x =Number of elements of a group of Z5 Z8x =Number of…

Q: Every cyclic group or order n is isomorphic to (Zn, +n) and every infinite cycle group is isomorphic…

A:

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: Suppose that 0: G G 5a group homomorphism. Show that 0 $(e) = 0(e) (ii) For every geG, (0(g))= 0(g)*…

A:

Q: Either give an example (with explanation), or explain why the following is not possible. (a) A…

A: According to the given information it is required to give an example that satisfies the given…

Q: G is a cyclic group of order15, then which is true a) G has a subgroup of order 4 b) G has a…

A:

Q: There are two group of order 4, namely Z, and Z2 Z2, and only one of them can be isomorphic to G/Z.…

A:

Q: The cyclic group of order 12 acts on {1,2,..., 12} with the following cycle structure. (1)…

A: Given that,  the group of order 12 In this case of necklace, there is no difference between…

Q: Let G = A4, the alternating group of degree 4. (a) How many elements of order 3 does G have? (b)…

A:

Q: 1+2n Prove that if (Q-(0},) is a group, and H = a n, m e Z} 1+2m is a subset of Q-{0}, then prove…

A:

Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).

A:

Q: : Show that if G is a cyclic group then G is abelian. : Let n>0 be a positive integer, and let a and…

A:

Q: True or false? The group S3 under function composition ◦ is not a cyclic group

A:

Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

A:

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

A:

Q: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

A:

Q: Let Dg be the Dihedral group of order 8. Prove that Aut(D8) = D8.

A: We have to solve given problem:

Q: 6. Let G be GL(2, R), the general linear group of order 2 over R under multiplication. List the…

A:

Q: Consider the group G = Dg × Z/6Z, and let N = Dg x {0}. Show that N 4 G and that G/N = Z/6Z. Now…

A: Given is a group G and a subgroup N of G. First we use the definition of Normal subgroup to show…

Q: (a) Draw the lattice of subgroups of Z/6Z. (b) Repeat the above for the group S3.

A:

Q: Let (G, -) be an abelian group with identity element e Let H = {a E G| a · a · a·a = e} Prove that H…

A: To show H is subgroup of G, we have show identity, closure and inverse property for H.

Q: 5. Let H and K be normal subgroups of a group G such that H nK = {1}. Show that hk = kh for all h e…

A:

Q: 6. (b) For each normal subgroup H of Dg, find the isomorphism type of its corresponding quotient…

A: First consider the trivial normal subgroup D8. The quotient group D8D8=D8 and hence it is isomorphic…

Q: Q3: (A) Prove that 1. There is no simple group of order 200. 2. Every group of index 2 is normal.

A: Sol1:- Let G be a group of order 200  i.e O(G) = 200                = 5² × 8. G contains k Sylows…

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = -S, \S| = 3 and (S) =…

A:

Q: 1. Let G be a cyclic group of order 6. How many of its elements generate G?

A: Any finite cyclic group of order 'n' has total ϕ(n) number of generators.  where 'ϕ' represents…

Q: 4. Let (G, *) be a group of order 231 = 3 × 7 × 11 and H€ Syl₁₁(G), KE Syl, (G). Prove that (a). HG…

A: The Sylow theorems are a fixed of theorems named after the Norwegian mathematician Peter Ludwig…

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: Find a non-trivial, proper normal subgroup of the dihedral group Dn-

A:

Q: Let Hand K be subgroups of an Abelian group. If |H| that HN Kis cyclic. Does your proof generalize…

A: This question is related to group theory. Solution is given as

Q: Let G be a group and let x EG of order 23. Prove that there exist an element z EG such that z6 – x.

A:

Q: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

A:

Q: Let G be a group, and define N to be the cyclic group N = ⟨xyx−1y−1 | x,y ∈ G⟩. (a) Prove N is a…

A:

Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

A:

Q: a. Show that (Q\{0}, + ) is an abelian (commutative) group where is defined as a•b= ab b. Find all…

A:

Q: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an abelian…

A:

Q: 2. (a) Let be a subgroup of the center of G. Show that if G/N is a cyclic group, then G must be…

A:

Q: 6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.

A:

Q: Show that every group G of order n is isomorphic to a subgroup of Sn. (This is also called Caley's…

A: