Mathematics MCQs for PPSC FPSC KKPSC BPSC SPSC NTS Test 8

Which of the following is a representation of S₃:

A). {< a, b > | a³ = b² = (ab)² = e}

B). {< a, b > | a² = b² = e }

C). {< a > | a² = e}

D). {< a, b > | a² = b² = (ab)² = e}

Let X = {1, 2, 3}. Then S₃ has ..................... elements:

A group all of whose elements are of finite order is called:

Product of two cyclic permutations is:

In S₄ group of permutation, number of even permutation is:

A permutation of degree n can be expressed as a product of:

Which of the following is a subgroup of S₃ = {a, a², b, ab, a²b, e}.

A). {e, a, a²}

B). {e, b}

C). {e, a², b}

D). All of These

A transposition is a cycle of length:

Let X has n elements. The set Sn of all permutations on X is a group w. r. to mappings:

If a group is neither periodic nor torsion free, then G is:

The A_n, set of all even permutations of S_n is a subgroup of S_n. Then order of A_n is:

A). n !

B). n!/3

C). n !/3

D). n + 1/2

If a homomorphism is also subjective the it is called:

A monomorphism is a homomorphism which is also:

A bijective homomorphism is called:

The only idempotent element in a group is:

Which of the following is Abelian:

A). S₂

B). S₃

C). S₄

D). S₅

A mapping Ø: G → G' is called homomorphism if for a, b ∈ G:

A). Ø (ab) = Ø (a) + Ø (b)

B). Ø (ab) = Ø (a) Ø (b)

C). Ø (ab) = Ø (a) - Ø  (b)

D). Ø (ab) = Ø (a) Ø (b)^-1

The relation of isomorphism between group is:

Let E be a group of even integers under binary operation of addition. Then which of the following isomorphic to E:

A group in which every element except the identity has infinite order is called:

In S_n number of odd permutation is:

A). n !

B). n!/3

C). n !/3

D). n + 1/2

A group G is Abelian. Then:

The group S_n is called:

Let D₄ = {<a, b>; a⁴ = b² = (ab)² = 1} be a dihedral group of order 8. Then which of the following is a subgroup of D₄.

A). {<a, b>; (ab)² = 1}

B). {<a, b>; a⁴ = b² = 1}

C). {<a³, b>; (a³b)² = 1}

D). {<a, b>; b² = 1}

Which of the following group with binary operation of ordinary addition is torsion free.