Can Subgroups Form a Group by Union Without Containing Each Other?

[Mathelogician]

Hi all,
Here i ask the fisrt serie of questions i couldn't solve;
A basic knowledge of group theory is supposed for solving them!
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1- Can you find 3 subgroups H, k and L of a group G such that H U k U L = G ;and no one of the 3 subgroups is a subgroup of the union of the other 2 subgroups { e.g., H is not a subgroup of (k U L) } ? ... [ In a simpler case assume that G is finite and |H|=|k|=|L|=|G|/2.]2- Is (R,+) finitely generated? Why?

3- let A,B<G and G be a finite group such that AB is not equal to AB. Then show that |G|>= |A|+|B|.------------------------------------------------------------
Thanks.

 

[MarkFL]

This topic is still under construction...

I have asked the OP to show some work or thoughts, and to create a new topic for the third question, so I ask that everyone wait until this has been completed before giving help.

Once the topics are completed, I will delete my post.

Thank you.