Does AB Equal G When |A| + |B| Exceeds |G|?

[joeblow]

If A and B are mere SUBSETS of G and |A| + |B| > |G|, then AB = G.

My thought is that a "weakest case" would be where A has |G|/2 + 1 elements (if |A| + |B| > |G|, then one of A or B must have over half of G's elements) and |G|/2 of them form a subgroup of G. Then taking B= the subgroup, it is true that AB = G because that one stray element when multiplied with the subgroup gives the rest of G. So, since AB = G in this case, it must be true in general.

Is this valid, or should I be doing something else?

 

[factor]

I'm wondering exactly why |G|/2 of them form a subgroup of G. By Lagrange's theorem the order of any subgroup must divide the order of the group, but suppose your group has order p where p is a prime, then it clearly can't have a subgroup of order 2 unless p is in fact equal to 2 in which case your subgroup satisfying the stated property would simply be the identity which is trivial.

 

[joeblow]

I don't mean to imply that any group will have a subgroup that has half the elements; I'm saying that *IF* G had a subgroup H with half of its elements, then choosing A = {H plus one element from G-H} and B = H, this proposition would hold. I think that such a case is the "weakest" case possible, so if it works for this case, it must be true in general. I think this because the closure property for subgroups allows for fewer elements to be introduced by elements from A being multiplied with elements of B.

Do you think that this is true?

 

[Useful nucleus]

When you add that stray element to H to form the set A, you actually got the whole group. Since the order of a group is always divisible by the order the subgroup, you cannot have a subgroup of G which has an order > |G|/2 unless this subgroup is the whole group.

This means that it is impossiple to have |A| + |B| > |G|, unless one of the subgoups is the whole group G.

 

[joeblow]

A and B are not necessarily subgroups of G; they are just subsets of G. So when I multiply A and B, I get H U aH = G. (where a is the "stray" element)

Since it works for this case, *does* it work for all cases? Is what I have a valid proof of the proposition?

 

[Office_Shredder]

You don't need a subset of A or B to be a group: If you just have |G|/2+1 elements as a subset of G, the group they generate needs to be G. It doesn't matter if they or some subset of them formed a subgroup to begin with or not