TIFR-2015-Maths-B-8

Question

For a group $G$, let $F(G)$ denote the collection of all subgroups of $G$. Which one of the following situations can occur ?

• A. $G$ is finite but $F(G)$ is infinite.
• B. $G$ is infinite but $F(G)$ is finite.
• C. $G$ is countable but  $F(G)$ is uncountable.
• D. $G$ is uncountable but $F(G)$ is countable.

Answer 1

If G is finite then F(g)= 2^finite is finte.

G is (infinite then F(g)= 2^(infinite) is infinte.

If G is countable(infinite) then F(g)= 2^{countable(infinite)} is uncountable.

G is uncountable then F(g)= 2^uncountable is uncountable.

So option c is correct.

Comments

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,

Answer 2

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.