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

Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is _____.

Let G be an arbitrary group. Consider the following relations on G :

R₁: ∀a,b ∈ G, aR₁b if and only if ∃g ∈ G such that a = g^-1bg

R₂: ∀a,b ∈ G, aR₂b if and only if a = b^-1

Which of the above is/are equivalence relation/relations?

Let U = {1, 2 ,..., n}. Let A = {(x, X) | x ∈ X, X ⊆ U}. Consider the following two statements on |A|.

I. |A| = n2^n–1

II. |A| = $$\sum\limits_{k = 1}^n {k\left( {\matrix{ n \cr k \cr } } \right)} $$

Which of the above statements is/are TRUE?