4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D

Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is

A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…

Q: - Show that the following subset is a subgroup. H = {o e S, l0(n) = n} S,

A:

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: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345

A: For group, 2*3*4=(2*3)*4.

Q: 28. Is every group a cyclic? Why?

A:

Q: Q/ How many non-trivial subgroups in s, ? a) 2 b) 3 c) 4

A: S3 is the set of permutations on the set {1, 2, 3}S3={e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2) }The…

Q: The following is a Cayley table for a group G. The order of 4 is: 2 3 5 2 3 4 5 3 4 1 2 4 2 1 3 2 3…

A: According to our company's guidelines I can only answer first question since you have asked multiple…

Q: · In a group, prove that (ab) = b-'a-|

A: As you asking for question number 7 , I solve for you.

Q: Are groups Z×10and Z×12 isomorphic

A: Concept:

Q: 11. Prove that every Cayley table is a Latin square for a group. That is, each element of the group…

A: To prove, each element of the group appears exactly once in each row and each column of a Cayley…

Q: (H,*) is called a of (G,*) if (H,*) is a group.

A:

Q: If a is an element of order 8 of a group G, and = ,then one of the following is a possible value of…

A: Given that a is an element of order 8 and a4=ak

Q: O In a group (G,x), if IGl= 37, the number of Passible Subgroups in G are

A: Note: Since you haven't mentioned which question you would like to get answered. We are providing…

Q: Consider Lof order 4 of the dihedral group a normal subaroup H. Duz then prove that for each hC H.…

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: In a group G,let a,b and ab have order 2.show that ab=ba

A:

Q: Give two reasons why the set of odd permutations in Sn is not a subgroup.

A: To show that, The set of odd permutations in Sn is not a subgroup. A subset H of the group G is a…

Q: Suppose that H is a subgroup of a group G and |H| = 10. If abelongs to G and a6 belongs to H, what…

A: Given: H is a subgroup of a group G and |H| = 10 To find: If a belongs to G and a6 belongs to H,…

Q: 5. If H = 122Z and K = 8Z are subgroups of (Z, +). Then H + K = ... %3D

A:

Q: Z, x is not a group.

A:

Q: Find all inclusion between subgroups in Z/48Z

A:

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: Suppose the Cayley table for B = {e, a, b, c}unde the binary operation * is given by * e b с a e e а…

A:

Q: 3. Consider the group (Z,*) where a * b = a + b – 1. Is this group cyclic?

A: 3. Given the group ℤ,* where a*b=a+b-1. Then, 1*x=x*1=x+1-1=x Here 1 serves as the identity for Z.

Q: 15. Suppose that N and M are two normal subgroups of a group G and that NO M = {e}. Show that for…

A:

Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

A:

Q: The following is a Cayley table for a group G. 2*5 4 = 1 1 3 1 4 2 4 4. 4. 5. 4. ENG pe here to…

A: 1

Q: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

Q: How many subgroups in Z/30Z 4 30 6 CO

A: Here we have group z/30z and find its number of subgroups.

Q: Suppose that G is a cyclic group and that 6 divides |G|. How manyelements of order 6 does G have? If…

A:

Q: How many subgroups are there in Z/14Z?

A: Subgroup: Let (G, ∙) be a group and S be a non‐empty subset of G.Then, S is called a subgroup of G…

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

A: Concept:

Q: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.

A:

Q: 10) Which of the following is not a group? * (Z,*), a* b = a + 2b Va, b E Z O (Z,*), a* b = a +b Va,…

A: Option (1) is correct.

Q: 13. Find groups that contain elements a and b such that |al = \b| = 2 and a. Jab| = 3, b. Jab| = 4,…

A:

Q: Suppose H and K are subgroups of a group G.  If |H|=12 and |K| = 35, find |H intersected with K|.…

A:

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: 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: ng to a group. If |a| = 12, |6| = 22, and (a) N (b) # {e}, prove that a® = b'1.

A:

Q: If a is an element of order 8 of a group G, and 4 = ,then one of the following is a possible value…

A:

Q: How many subgroups does Z/60Z have?

A:

Q: Let a,B ES ( Symmetric group) where a = (1,8,5,7)(2,4) and B=(1,3,2,5,8,4,7,6)- Compute aB. Attach…

A:

Q: Consider the discrete group G of order 8 that has the following Cayley diagram e If we have the…

A: The sequence of operations is fcagec. Each element g of G is assigned a vertex: the vertex set…

Q: Exercise 5 Is every group f ordler u 6) n< 60, (c) n=300 solvable (a) n=1000,

A:

Q: Q/ How many non-trivial subgroups in s, ?

A: For the given statement

Q: Any group of order 520 is simple

A: Simple group:     "A simple group is a nontrivial group whose only normal subgroups are the trivial…

Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

A:

Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

Q: (5) Show that in a group G of odd order, the equation x² = a has a unique solution for all a e G.

A:

Q: Consider the "clock arithmetic" group (Z12,O), together with subgroup H= {0,4, 8} Write all cosets…

A: Consider the ‘clock arithmetic’ group Z12, ⊕, together with subgroup H=0,4,8. The objective is to…

Q: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

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

A:

Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…

A:

Q: Consider a normal subqroup H of order 4 of the dihedral group b2-lab)? => ther Dus< a, b: Q= n prove…

A: Write the general formula for the element of the group and use operation

Q: In Z24 the number of all subgroups is 8 O 6.

A: We are asked to find the subgroup of z24