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

In summary, the conversation is about a request for help in solving three questions related to group theory. The first question asks for three subgroups of a group G that can form the entire group without any one subgroup being a subgroup of the union of the other two. The second question asks if the group (R,+) is finitely generated. The third question is still under construction and will be posted in a separate topic. The OP has been asked to show some work or thoughts before receiving help.
  • #1
Mathelogician
35
0
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!
------------------------------------------------------------

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.
 
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  • #2
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.
 

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