7. You have previously proved that the intersection of two subgroups of a group G is always a sub- group. For G = S3, show that the union of two subgroups may not be a subgroup by providing a %3D counterexample.

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

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

A:

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

A: To determine the possible subgroups H satisfying the given conditions

Q: Every

A: We will be using sylow's theorems and it's consequences to arrive at the conclusion that statement…

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

A:

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: 11. Find the cyclic subgroup of D4 generated by µp². What is the order of this subgroup?

A:

Q: Explain why S8 contains subgroups isomorphic to Z15, U(16), and D8.

A:

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: 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: When we say xH = Hx where H is a normal subgroup of G and x is an element of G, what exactly does…

A:

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: 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: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.

A:

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: 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: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

A: Given question:   Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

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

A:

Q: (3) Suppose n= |T(x)| and d=|x| are both finite. Then, using fact 3 about powers in finite cyclic…

A:

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: The subgroup {e} is called the nontrivial, that is, a subgroup that is not e is nontrivial.…

A:

Q: In (Z10, +10) the cyclic subgroup generated by 2 is (0,2,4,6,8). True False If G = {-i,i,-1,1} be a…

A:

Q: 3) Show that the subgroup of Dg is isomorphic to V4.

A:

Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

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: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

Q: If G is an infinite group, what can you say about the number ofelements of order 8 in the group?…

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: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

A:

Q: 2*. Let Q/Z be the group described in problem 12 of Worksheet 1.1. Find list the elements of the…

A: To identify the subgroups generated by the given elements in the quotient group Q/Z.

Q: If H and K are subgroups of G, Show tht H intersecting with K is a subgroup of G.  (Can you see that…

A: Use the 2-step subgroup test to prove H Ո K is a subgroup, which states that,

Q: I'm unsure of how to approach this... Let N be a finite group and let H be a subgroup of N. If |H|…

A: According to the given information, Let N be a finite group and H be a subgroup of N.

Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

A:

Q: What is the relationship between a Sylow 2-subgroup of S4 and the symmetry group of the square? that…

A:

Q: 8. Is Zg isomorphic to D4? What about Z4 and D4? Can you find a subgroup of D4 isomorphic to Z4?

A: Now we have to answer the above question .

Q: For G = S3, show that the union of two subgroups may not be a subgroup by providing a…

A: PROOF

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: 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: 2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

A: Here we have to show that given groups are isomorphic

Q: What could the order of the subgroup of the group of order G| = 554407

A: We find the possible order of all the subgroups of the group G, where |G|=55440 by using Lagrange's…

Q: How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?

A:

Q: Suppose that G is cyclic and G = (a) where Ja| = 20. How many subgroups does G have?

A:

Q: Let 0:Z50-Z15 be a group homomorphism with 0(x)=4x. Then, Ker(Ø)= {0, 10, 20, 30, 40)

A:

Q: The group (Z, t6) contains only 4 subgroups

A:

Q: Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x…

A: Let X,◊ and Y,* are two groups. The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.…