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

1. $G$ is finite but $F(G)$ is infinite.
2. $G$ is infinite but $F(G)$ is finite.
3. $G$ is countable but  $F(G)$ is uncountable.
4. $G$ is uncountable but $F(G)$ is countable.
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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.

by Veteran (62.5k points)
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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.

by Loyal (9.9k points)