Question

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

Answer 1

Answer: The size of a largest possible proper subgroup of g is 42.

Step-by-step explanation:

Let G be a group such that |G|=84

Let H be the largest proper subgroup of G, then the order of H divides the order of G.

[∵ the order of any subgroup of G divides the order of G]

So the possible order of H= 84,42,...

As H is the largest proper subgroup then |H|=42

Since a proper subgroup of a group G is a subgroup H which is a proper subset of G (i. e. H ≠ G). or |H|≠|G|

Hence, The size of a largest possible proper subgroup of g is 42.