9. Prove that H ne Z} is a cyclic subgroup of GL2(R). . Subgraup chésed in Pg 34

Q: Suppose that a group G of order 231 has a normal subgroup N of order 11. Then, G/N is cyclic O False…

A: Given that G is a group of order 231 and N is an normal sub-group of G of order 11. To show: G/N is…

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

A:

Q: compute the 3 -sylow subgroups of S5

A: n3 (S5 ) =(1+3k) /40

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

A:

Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

A:

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

A:

Q: 3. Prove that (Z/7Z)* is a cyclic group by finding a generator.

A: Using trial and error method, seek for an element of order 6.

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: Find a noncyclic subgroup of order 4 in U(40).

A: Let U(40) be a group. Definition of U(n): The set U(n) is set of all positive integer less than n…

Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

A:

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: Show that no proper subgroup of S4 contains both (1,2,3, 4) and (1,2).

A: We have to show that no proper subgroup of S4 contains both 1,2,3,4 and 1,2. We know that the…

Q: is a subgroup of Z1, of order: 3 12 O 1 The following is a Cayley table for a group G. 2. 3.4 = 2 3…

A:

Q: Suppose that H is a subgroup of Z under addition and that H contains 250 and 350. What are the…

A: Given H be a subgroup of (Z,+) containing250 and 350Note that,  GCD(250 , 350)=1⇒by property of GCD,…

Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: 16* Find an explicit epimorphism from S5 onto a group of order 2

A: To construct an explicit homomorphism from S5 (the symmetric group on 5 symbols) which is onto the…

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

A:

Q: c) Show that Z,,+, is a cyclic group generated by 3

A: 3(c) To check if 3 is generator of (Z5 , +5) , we must check that 3 generates all the members of Z5…

Q: 17. Show that every group of order (35)° has a normal subgroup of order 125.

A:

Q: Let Ø: Z50 → Z,5 be a group homomorphism with Ø(x) = 4x. Ø-1(4) = %3D O None of the choices O (0,…

A: Here we will find out the required value.

Q: Show that ( Z,,+,) is a cyclic group generated by 3.

A:

Q: prove That :- let H and K be subgroups of agroupG of the m is ormal a HK is subgroup of G →1f one

A: Subgroup of a group G

Q: 1. Prove that a subgroup which is generated by W-marginal subgroups is itself W-marginal.

A:

Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

A:

Q: 5. How many automorphisms does Klein's 4-group have?

A: No of automorphism of Klein's 4-group:  K4={e,a,b,c} f1=eabceabc = I f2=eabceacb = bc f3=eabcecba =…

Q: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

Q: Show that the subgroup generated by any two distinct elements of order 2 in S3 is all of S3.

A: Given that, S3 is a symmetric group of permutations. Thus, S3 has 6 elements. By using Lagrange's…

Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

Q: c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

A: #Dear user there is a mistake in the question the assumption is for the element c and d of a group…

Q: 4-Let (Z12, +12) be a group and let S={4,6}, find subgroup H generated by S. if exist

A: I have used the definition of subgroup generated by a subset.

Q: Show that every group of order (35)3 has a normal subgroup of order 125

A: Given, A Sylow 5-subgroup of a group of order 353 is of order 125. The divisors of 353 that are not…

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: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: Q1/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*)

A:

Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

A:

Q: Let (Z12, +12) be a group , if we take {0,4,8} for the set H then ({0,4,6}, +12) is evidently a…

A: Let H=0, 4, 6 We know that the operation in ℤ12 is addition. So, the element of left coset is of the…

Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

A:

Q: (a) Compute the list of subgroups of the group Z/45Z and draw the lattice of subgroups. (prove that…

A: In the given question we have to write all the subgroup of the group ℤ45ℤ and also draw the the…

Q: Q2/ In (Z9, +9) find the cyclic subgroup generated by 1,2,5

A:

Q: Show that the only Normal subgroup of S3(All permutations of 3 distinct elements) is the subgroup…

A:

Q: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

A:

Q: If H and K are subgroups of G of order 75 and 242 respectively, what can you say about H N K?

A: Solution

Q: Obtain al the Sylow p-subgroups of (Z/2Z) X S3

A:

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

A: Follow the steps.

Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

Q: i need help with attached question for abstract algebra please

A:

Q: 7. You have previously proved that the intersection of two subgroups of a group G is always a sub-…

A:

Q: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

Q: Show that a group of order 77 is cyclic.

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.