Question 1 |

Which of the following statements is/are TRUE for a group G?

**MSQ**If for all x,y \in G, (xy)^2=x^2y^2, then G is commutative. | |

If for all x \in G, x^2=1, then G is commutative. Here, 1 is the identity element of G. | |

If the order of G is 2 , then G is commutative. | |

If G is commutative, then a subgroup of G need not be commutative. |

Question 1 Explanation:

Question 2 |

Let G be a group of order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?

Both G and H are always cyclic. | |

G may not be cyclic, but H is always cyclic. | |

G is always cyclic, but H may not be cyclic. | |

Both G and H may not be cyclic. |

Question 2 Explanation:

Question 3 |

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is _______.

34 | |

35 | |

5 | |

7 |

Question 3 Explanation:

Question 4 |

(G,*) is an abelian group. Then

x=x^{-1} for any x belonging to G | |

x=x^{2} for any x belonging to G | |

(x * y)^{2}=x^{2} * y^{2} , for any x,y belonging to G | |

G is of finite order |

Question 4 Explanation:

Question 5 |

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is ________.

42 | |

21 | |

24 | |

84 |

Question 5 Explanation:

Question 6 |

f (G,.) is a group such that (ab)^{-1}=a^{-1}b^{-1},\forall a,b \in G, then G is a/an

Commutative semi group | |

Abelian group | |

Non-abelian group | |

None of these |

Question 6 Explanation:

Question 7 |

There are two elements x,y in a group (G,*) such that every element in the group can be
written as a product of some number of x's and y's in some order. It is known that

x * x = y * y = x * y *x * y = y* x * y *x = e

where e is the identity element. The maximum number of elements in such a group is _______.

x * x = y * y = x * y *x * y = y* x * y *x = e

where e is the identity element. The maximum number of elements in such a group is _______.

3 | |

4 | |

5 | |

6 |

Question 7 Explanation:

Question 8 |

Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L\neqG and that the size of L is at least 4. The size of L is _______.

4 | |

5 | |

6 | |

7 |

Question 8 Explanation:

Question 9 |

The arithmetic mean of attendance of 49 students of class A is 40% and that of 53 students of class B is 35%. Then the percentage of arithmetic mean of attendance of class A and B is

27.20% | |

50.25% | |

51.13% | |

37.40% |

Question 9 Explanation:

Question 10 |

Consider the set S = {1, \omega ,\omega ^{2}}, where \omega and \omega ^{2} are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

A group | |

A ring | |

An integral domain | |

A field |

Question 10 Explanation:

There are 10 questions to complete.

Please add a feature to BOOKMARK questions for revision. It would be really helpful.

check question 20 ,answer should be option c

For the Q24, The answer is option C