Mathematics III Exam Prep
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Mathematics III Exam Prep
English 21 leveled MCQs Free
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10 marks Long answer
State Lagrange's theorem and give an example in support of it.
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Lagrange's theorem states that the order of a subgroup divides the order of the group; for example, in \( S_3 \), the subgroup of order 2 divides the group of order 6.
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10 marks Long answer
Is every group of prime order cyclic? If it is, then give an example in the support of it.
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Yes, every group of prime order is cyclic; for example, the group of order 5 is cyclic and can be generated by any of its non-identity elements.
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10 marks Long answer
Show that the group ??????={1,?????? ??????2}, (?????? is the cube root of unity) is a cyclic, how many generators are there.
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The group of cube roots of unity is cyclic, and it has two generators.
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10 marks Long answer
Define a cyclic group and give two examples.
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A cyclic group is generated by a single element; examples include \( \mathbb{Z} \) under addition and \( \mathbb{Z}/n\mathbb{Z} \).
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10 marks Long answer
Define the quaternion group Q₈. Write its elements, show that it is non-abelian.
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The quaternion group Q₈ consists of {1, -1, i, -i, j, -j, k, -k} and is non-abelian because the multiplication is not commutative.
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10 marks Long answer
Prove that if ?????? is a generator of a cyclic group ?????? then ??????−1 is also a generator of G.
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If \( g \) is a generator of a cyclic group, then \( g^{-1} \) generates the group as it produces all elements in reverse order.
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10 marks Long answer
Define a ring and explain its different types with suitable examples.
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A ring is a set with two operations; types include commutative rings, like integers, and non-commutative rings, like matrices.
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4 marks Medium answer
Show that the set ??????={1,??????,??????2} of cube roots of unity forms an abelian group under multiplication of complex numbers.
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The set of cube roots of unity forms an abelian group as it satisfies closure, associativity, identity, and inverses.
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4 marks Medium answer
Define left coset and right coset of a subgroup H in a group G.
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A left coset is a set of the form {g*h | h ∈ H} and a right coset is {h*g | h ∈ H} for some g in G.
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4 marks Medium answer
Prove that every cyclic subgroup is an abelian group.
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Every cyclic subgroup is abelian because all elements can be expressed as powers of a single generator.
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4 marks Medium answer
Write the Cayley table of Klein's four-group K₄ = {e, a, b, c}.
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The Cayley table for K₄ shows the results of the group operation for all pairs of elements.
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4 marks Medium answer
Define permutation and solve any one example.
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A permutation is an arrangement of elements; for example, the permutation of {1,2,3} can be (2,1,3).
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4 marks Medium answer
Define a ring with one suitable example.
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A ring is a set with two operations; for example, the set of integers with standard addition and multiplication is a ring.
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4 marks Medium answer
Show that every subgroup of an abelian group is normal subgroup.
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In an abelian group, every subgroup is normal because the group operation is commutative.
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2 marks Short answer
Define a binary operation.
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A binary operation is a calculation that combines two elements to produce another element.
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2 marks Short answer
Define identity element.
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An identity element is an element that, when combined with any element of the set, leaves that element unchanged.
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2 marks Short answer
Define the order of a group.
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The order of a group is the number of elements in the group.
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2 marks Short answer
Define cyclic group.
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A cyclic group is a group that can be generated by a single element.
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2 marks Short answer
Define Ring.
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A ring is a set equipped with two binary operations satisfying properties similar to those of addition and multiplication.
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2 marks Short answer
Define field.
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A field is a set in which addition, subtraction, multiplication, and division are defined and behave as they do for rational numbers.
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2 marks Short answer
Define an abelian group.
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An abelian group is a group in which the operation is commutative.
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