Qi/ If (H,*) is collection of subgroups of (G,*) then (U H,*) is subgroup of (G,*) Q2/ In (Z9, +9) find the cyclic subgroup generated by 1, 2,5

Q: The elements of the quotient group (/18) are 1- 2- {.-208,-108,008,108,208,.} 3- (0,1,2,3, .8) 4-…

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Q: A group G is cyclic if and only if there exists at G such that G={a^|n  ∈ Z}. True or False then why

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Q: Explain Special Linear Group is a Normal Subgroup of General Linear Group?

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Q: QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a € G} and we…

A: As you have posted multiple questions, so according to our guidelines we will solve only first…

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will  proof the statement otherwise disprove we taking the…

Q: 3) Let G be a cyclic group then a) Z(G) = G,where Z(G)is the center of b) Every subgroup of G is a…

A: our objective is to show which options are true in the above questions.

Q: Let G a finite group, g E G such

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Q: Let G be a group and a ∈ G. Suppose that |a| = 18, find |a^21|. Suppose that |a| = 32, find a…

A: We are given that G is a group, and a ∈ G.A) The order of element a is given as 18. a=18 ⇒a18 = e,…

Q: 1- The index 2- infinite group 3- permute elements B prove that (cent G,*) is normal subgroup of the…

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Q: (a) (Z4, +). (b) (Z5, .). (c) (Z5, +). (d) (Z4, .). (e) (Z6, .).

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Q: (b) Let H= ((3,3, 6)), the cyclic subgroup of G generated by (3,3,6). Determine |G/H.

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Q: Consider the group G = {x E R such that x 0} under the binary operation x*y=-2xy O x*x*x=4x^3…

A: Multiplication of the elements of the group elements with respect to binary operation

Q: 2. If H and K are normal subgroups of a group G and H^K = {e}, prove that G is isomorphic to a…

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Q: Given that G is a group and H is a subgroup. What is the result of (b^-1)^-1 if b is an element of…

A: Given that     G is a group and H is a subgroup of G.    Inverse of an element:  Let G be a group…

Q: for the cyclic group of order 16 generated by a , show that a^4 generates an invariant subgroup

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Q: 5/ Let G be group of class p9 a Prime Setting that proves that actual Subgroup of G is a cyclie is a

A: We know that every group of prime order is cyclic

Q: find multiplication table of dihedral group D_4where rotations are denoted by r,r^2, r^3.…

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Q: Let G be a group and H a subgroup of G. H is a normal (or distinguished) subgroup of G if and only…

A: The Center of the group is defined as, C=x∈G | ∀y∈G yx=xy. The given definition 1' is: A subgroup H…

Q: Let G be a group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

Q: Find the order of the factor group U(16)/(9)where U(16)={1,3,5,7,9,11,13,15} with operation…

A: Since , we have to find order of the factor group U(16)/<9> Concept: Order of factor group…

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: 4. There is a theorem that says that every element g € GL(2, R) can be written, in a unique way, as…

A: Given:  g=053-12           We have to find k, a, n such that g=kan where k∈K, a∈A and n∈N.

Q: Let G and H be groups, and let 0 : G -» H be a {x E G 0(x) = e} is called the kernel of 0 and is…

A: From the given information:Let G and H be the groups

Q: If (G,*) is group and each subgroups of (G,*) is normal subgroups then (G,*) is abelian group.

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Q: Enter the smallest subgroup of M₂ (R)* containing the matrix -5 -3 7 4 as a set

A: Given: -5-374∈M2ℝ* To find: Smallest subgroup of M2ℝ* containing given element.

Q: Q/ Let the Cayley group C₁ (S) = C₂ Find the group and the subset S. (3)

A: Introduction: A Cayley graph is a graph that represents a group and its generators. Given 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: show rotations and reflections graphically using regular pentagon dihedral group elements…

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Q: QUESTION 6 Consider the groups U(8) and Z () Determine the identity element in the group U(8) x ZA…

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Q: prove or disprove The intersection of any two distinct left cosets in the group is empty set

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…