· In a group, prove that (ab) = b-'a-|

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

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: ii) Does there exist a group G such that G/[G,G] is non-abelian? Give an example, or prove

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Q: Theorem :- Let (G,-) be a group then :- 1- (Hom(G),) is semi group with identity. 2- (A(G),) is…

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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: Prove that a group of order 7is cyclic.

A: Solution:-

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

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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: Show that the translations are a normal subgroup of the affine group.

A: To show: The translations are a normal subgroup of the affine group

Q: Give an example, with justification, of an abelian group of rank 7 and with torsion group being…

A: consider the equation

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

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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: ab 2. Show that (Z, *) is an abelian (commutative) group, where' is defined as a*b = and Z is the…

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Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

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

A: Given that the quotient group Z8.⊗. We have to find the elements of the quotient group Z8.⊗.…

Q: PROVE |z.|-|z,|<|z, – z||

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Q: Determine all cyclic groups that have exactly two generators.

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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: chow that An s a Group with respect, to Cooplesition of functjon.

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Q: Show that a subgroup of a solvable group is solvable.

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Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: Show that every group of order < 60 is solvable.

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Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

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

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Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

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Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank&gt;1 has trivial center

Q: The order of identity element equal itself in any group

A: .

Q: Prove that An even permutation is group w.y.t compostin Compostin function.

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Q: prove that esch'x - In (* + x2 %3D

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Q: List six examples of non-Abelian groups of order 24.

A: The Oder is 24

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

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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: f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn

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Q: Determine the galois group

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Q: 2. Is the set Z3 = {0,1,2} form a group with respect to addition modulo 3 how about to…

A: We use caley's table to verify the properties of a group.

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: Show that there are two Abelian groups of order 108 that haveexactly one subgroup of order 3.

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Q: not

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Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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

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Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

Q: 1. Prove that free groups are torsion-free.

A: Free groups are torsion-free: It is the situation that the free group FX on a set X contains no…

Q: 8. Show that (Z,,×s) is a monoid. Is (Z.,×6) an abelian group? Justify your answer

A: Note: since you have posted multiple questions . As per our guidelines we are supposed to solve one…

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

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Q: Prove that le-2z| 0.

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Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.