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

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: Prove that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.

A: If G is the simple group of order 60 That is | G | =60. |G| = 22 (3)(5). By using theorem, For every…

Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

Q: Every

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

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: Express U(165) as an internal direct product of proper subgroups infour different ways

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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: 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…

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Q: If G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G with H…

A: Given: Group G=1,2,3,4,5,6 under multiplication modulo 7. Subgroup H=1,2,4 of G. To find:   a) Right…

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: Z, x is not a group.

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Q: List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that…

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Q: 3. Define Lie group.

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Q: The intersection of two normal subgroups is always normal sub group. (True/False)

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Q: G group is called meta- Abelian if it has an Abelian subgroup N which is normal in G G and is…

A: (Solving the first question as per our guidelines) Question: Show that S3 is meta-Abelian.

Q: (3) Show that 2Z is isomorphic to Z. Conclude that a group can be isomorphic to one of its proper…

A: (2ℤ , +) is  isomorphic to (ℤ , +) .              Define f :(ℤ , +) →(2ℤ , +) by…

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

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Q: What is the kernel of homomorphism of \phi:Z4->Z10 defined by \phi(x)=5x?

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Q: find Aut(Z30).  Use the FUndamental Theorem of Abelian Groups to express this group as an external…

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

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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: Sz,0) be a permutation group. Then all elements in One to one, onto function. Onto function.

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

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

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Q: Find the number of G-orbits in X := {1,2,3, 4, 5, 6, 7, 8, 9, 10}, where G < S10 is the subgroup…

A: Solution:Let g1 = (1,2), g2 = (3,4,5) - and g3 = (6,7,8,9). All three cycles at sight are…

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

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Q: The group Cs3,0) is normal group solvable ?

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

A: H is normal subgroup of G. And a belongs to G.  O( aH) = 3 in G/H and O(aH) in G/H divides  O(a) in…

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: Write U(32) as the internal direct product of two proper subgroups.

A: Given:         U32 We use the theorem namely Ut≈Usst   because Usst is a subgroup of Ust

Q: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

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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: Prove that the alternating group is a group with respect to the composition of functions?

A: Sn is the set of all permutations of elements from 1,2,.....,n which is known as the symmetric group…

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

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Q: Find the group homomorphism between (Z, +) and (R- (0},.)

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Q: The Kernal of any group homomorphism is normal subgroup True False

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Q: An is a * normal subgroup of Sn None of the choices not a subgroup of Sn subgroup of Sn but not…

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Q: t subgroups and quotient groups of a solvable group are solvable.

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Q: Decide whether (Z, -) forms a group where : Z xZ Z (a) is the usual operation of subtraction, i.e.…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Let let G₁ be A be of Suppose Subgroup index a group and a normal of finite G+₁ that H

A: We know that if G is a group and H is a subgroup of G and x is an element in G of finite order n. If…

Q: 15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the…

A: To construct a homomorphism from Z24 , which is onto a group of order 6.

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

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Q: 1- (Z,+) is a subgroup of (Q,+) 2- (Q,+) is a a subgroup of (R, +) 3- (R,+) is a a subgroup of (C,+)…

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

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

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