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Test: Group Theory - 5 - Mathematics MCQ


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20 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Test: Group Theory - 5

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Test: Group Theory - 5 - Question 1

The inverse of an even permutation is​

Test: Group Theory - 5 - Question 2

The order of the permutation  is

Detailed Solution for Test: Group Theory - 5 - Question 2

By the theorem, the order of any permutation ζ in Sn is equal to the l.c.m. of the orders of the disjoint cycles in ζ. Here, = (1 2 4 5)(3 6).
The order of the permutation is l.c.m. of (1 2 4 5) and (3 6).
l.c.m. (4, 2) = 4
as o(1 2 4 5) = 4 and o(3, 6) 

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Test: Group Theory - 5 - Question 3

A element aP of a finite cyclic group G of order n is a generator of G iff 0 < p < n and also

Test: Group Theory - 5 - Question 4

Suppose Km = {P∈Sm|, |P| is odd prime}. Determine the set for which m ≥ 3 Km a subgroup of Sm.

Detailed Solution for Test: Group Theory - 5 - Question 4

Since Km is a subset of Sm, then the set will be {3, 5, 7, 11, 13, …}.

Test: Group Theory - 5 - Question 5

If a ∈ G is of order n and P is prime to n, then the order of aP is

Test: Group Theory - 5 - Question 6

The generators of the group G = {a, a2, a3, a4 = e} are

Detailed Solution for Test: Group Theory - 5 - Question 6

we have o(G) = 4 and prime to 4 are 1 and 3

Test: Group Theory - 5 - Question 7

If order of group G is P2, where P is prime, then

Test: Group Theory - 5 - Question 8

A relation (34 × 78) × 57 = 34 × (78 × 57) can have __________ property.

Detailed Solution for Test: Group Theory - 5 - Question 8

For any three elements(numbers) a, b and c associative property describes (a ×  b) × c  =  a × (b  × c) [for multiplication]. Hence associative property is true for multiplication and it is true for multiplication also.

Test: Group Theory - 5 - Question 9

The inverse of an odd permutati on is

Test: Group Theory - 5 - Question 10

Let R be the ring of all 2 × 2 matrices with integer entries. Which of the following subsets of R is an integral domain?

Detailed Solution for Test: Group Theory - 5 - Question 10

Let

Let

⇒ AB = BA ∀ A, B ∈ R

⇒ R1 is commutative if AB = 0
⇒ A = 0 or B = 0
⇒ R1 has no zero divisors

Test: Group Theory - 5 - Question 11

Statement: All cyclic groups are abelian. Statement B: The order of cyclic group is same as the order of its generator.

Test: Group Theory - 5 - Question 12

If f = (2 3) and g = (4 5) be two permutation on five symbols 1, 2, 3,4, 5 then gf is

Test: Group Theory - 5 - Question 13

The permutation  is equal to

Detailed Solution for Test: Group Theory - 5 - Question 13

we have = (135) (24) = (13) (15) (24)

Test: Group Theory - 5 - Question 14

Statement A : Every isomorphic image of a cyclic group is cyclic.
Statement B : Every homomorphic image of a cyclic group is cyclic

Detailed Solution for Test: Group Theory - 5 - Question 14

Correct option is D. Both the statements are true.

Test: Group Theory - 5 - Question 15

The idempotent element in a group are

Detailed Solution for Test: Group Theory - 5 - Question 15

Every group has exactly one idempotent element : the identity.

Test: Group Theory - 5 - Question 16

If number of left cosets of H in G are n and the number of right cosets of H in G are m, then

Test: Group Theory - 5 - Question 17

Given, permutation  is equivalent to

Test: Group Theory - 5 - Question 18

Every group of prime order is

Test: Group Theory - 5 - Question 19

Given, the permutation C = ( 1 2 3 4 5 6 7) then C3 is

Detailed Solution for Test: Group Theory - 5 - Question 19

 we have C = (1234567) = 
which implies C2 = 
which implies C3
= 1 4 7 3 6 2 5 
Hence C3 = (1 4 7 3 6 2 5)

Test: Group Theory - 5 - Question 20

If H1 and H2 are two right coset sets of Subgroup H1, then

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