Quiz for II M. Sc

Let G be a group of order 77. Then the centre of G is isomorphic to

Let G be a cyclic group of order 848. Which of the following are possible orders of quotient groups of G?

• 28

• 16

• 53

• 54

Choose all the correct statements

• A group of order 2023 can not have an element of order 7

• A group of order 2023 can not have an element of order 19

• A cyclic group of order 2023 has exactly one subgroup of order 17

• A cyclic group of order 2023 has a unique subgroup of order 23

• For every positive integer n, there exists a group which contain exactly n elements

• For every positive integer n, there exists a non-abelian group which contains exactly n elements

• I and J are fields

• I is a field and J is not a field

• I is not a field but J is a field

• I and J are not integral domains

Let H = {e, (1, 2), (3, 4)} and H = {e, (1, 2), (3, 4), (1, 3), (2, 4), (1, 4), (2, 3)} be subgroups of symmetric group of order 4, where e denotes the identity element of symmetric group of order 4. Then which of the following are correct?

• a unique solution in R

• N number of distinct solution in R

• no solution in any interval containing 0 for some P

• a unique solution in an interval containing 0

• has a unique solution

• a family of straight lines as characteristics

• solution which vanishes at (2, 1)

• infinitely many solutions

Let X be a connected subset of real numbers. If every element of X is irrational, then the cardinality of X is

• infinite

• countably infinite

• 2

• 1