For a group $G$, let $F(G)$ denote the collection of all subgroups of $G$. Which one of the following situations can occur ?
If G is finite then F(g)= 2finite is finte.
G is (infinite then F(g)= 2(infinite) is infinte.
If G is countable(infinite) then F(g)= 2countable(infinite) is uncountable.
G is uncountable then F(g)= 2uncountable is uncountable.
So option c is correct.
is the total number of subgroups of a group is O(2n)..because of langrage's theorem there must be a less than subgroups of a group,
If order of a group is suppose N
Number of subgroup can be possible=2^N
N--->infinite then 2^N----->infinte
Use this logic .
And we know that there are two types infinite.
Infinite type are:::(1)Countable suppose to be infinite (2) Uncountable is infinite
Apply above logic and u will get option C is right option.