Group Theory

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

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?

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

(G,*) is an abelian group. Then

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

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

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 _______.

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 _______.

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

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