Give the example that group A is not null and -00 = inf (A) and (sup (A) = max (A).

Q: 9. Prove that a group of order 3 must be cyclic.

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Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: Let G be a group containing elements a and b. Express (ab)^2 without parentheses. Do not assume that…

A: Given that G is a group and let a,b belongs to G. The given expression is

Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

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Q: nilpotent

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Q: Prove that a group of order 12 must have an element of order 2.

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Q: The group GLQ,R) abelian group is an

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Q: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Find an isomorphism from the group G = to the multiplicative group {1, i, – 1, – i} in Example 3 of…

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Q: For any elements a and b from a group and any integer n, prove that(a-1ba)n = a-1bna.

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Q: Let G = : a – b = c – d, a,b, c, d E R Show that G is a group under (the usual) matrix addition.

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Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

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Q: (a) What does it mean for two groups to be isomorphic?

A: see my solution below

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: Prove that a group of order 15 is cyclic

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Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: Show that any group of order 4 or less is abelian

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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: If a is an element of order 8 of a group G,

A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e  where e is an identity…

Q: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

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Q: Prove that in a group, (a-1)-1 = a for all a.

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

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

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Q: Q2: If G = R- {0} and a * b = 4ab ,show that (G,*) forms a commutative group? %3D

A: To show for the commutative group of (G, *), we verify the following properties of the commutative…

Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

A:

Q: order 8 of a group G, and =

A: Given that order of a is 8 .Then  a8=e Rearrange  a little bit , we can have a42=e Hence order of…

Q: 3. Show that G = {a+bv2 |a,b e Q} is a group under the usual addition.

A: 3. Let G is said to be group under binary operation (.). It should satisfy the following properties.…

Q: State the first isomorphism theorem for groups and use it to show that the groups/mz and Zm are…

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Q: Prove that there is no simple group of order 210 = 2 . 3 . 5 . 7.

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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: (i). There is a simple group of order 2021.

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Q: Given two examples of finite abelian groups

A: Require examples of finite abelian groups.

Q: Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

A: We know that if two groups are isomorphic than they have same number of elements i.e. their…

Q: Show that the set S = (1, i, - 1, -0 is an abelian group with respect to the multiplication

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Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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Q: Consider the set S of ordered pairs of real numbers together with the operation defined by (a, b)*…

A: As per the company rule, we are supposed to solve one problem from a set of multiple problems.…

Q: Let G be a group. V a, b, c d and x in G, if axb cxd then ab = cd then G is necessa: Abelian Of…

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Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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Q: List all elements of U(10) and give a multiplication table for the group U(10)

A: Given  that, The group U(10). We have to find the all elements in U(10) and multiplication table for…

Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…

A: An automorphism of a group is the permutation of the group which preserves the property  ϕgh=ϕgϕh…

Q: Let (G1, ) and (G2, *) be two groups and p: G1 G2 be an isomorphism. The

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Q: inf (A) Give the example that group A is not null and -00 = and (sup (A) = max (A). %3D <י(S

A: Let  A=-n where n∈N  A is not null inf(A)=-∞sup(A)=-1=max(A)

Q: Write out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In…

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Q: Find the order of each element of the group Z/12Z under addition

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Q: The inverse of 3 in the group (Z5, o5) is

A: Definition:

Q: Let a,b be elements of S6 (symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).verify that…

A: Given: Let a,b be elements of S6 (Symmetric group) where a=(1,2)(4,5) and b=(1,6,5,3,2).

Q: Every abelian cy elic O True O False group is

A: to check  whether every abelian group is cyclic or not? proof let a euler group U8=1,3,5,7 let…

Q: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

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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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