Proving A Subset of Finite Group G

[ehrenfest]

[SOLVED] Larson 4.4.13

Homework Statement

A is a subset of a finite group G, and A contains more than one-half of the elements of G. Prove that each element of G is the product of two elements of A.

Homework Equations

The Attempt at a Solution

Is that even true? What if G is just the union of the cyclic group with 20 elements and the cyclic group with 21 elements. Let A = C_21. ord(G) = 20+21-1=40. A has more than half of the elements of G but you cannot get any elements of the C_20 subgroup except the identity with a product of elements of the C_12 subgroup.

 

[morphism]

The union of two groups? What binary operation are you giving it, i.e. how is C_20$\cup$C_21 a group?

 

[ehrenfest]

Oh. I forgot that the binary operation has to be defined between elements of C_20 and C_21, not just within C_20 and within C_21.

Then it probably is true. Let me think about it.

 

[Dick]

Oh. I forgot that the binary operation has to be defined between elements of C_20 and C_21, not just within C_20 and within C_21.

Then it probably is true. Let me think about it.

Do that. This is not the most difficult question you've posted by a long shot.

 

[ehrenfest]

I assume that the two elements in the problem statement are not necessarily distinct. Otherwise, Z_11 and A = {0,1,2,3,4,5} is a counterexample because you cannot add any two distinct elements of A to get 10.

Let S be the set of all ordered 2-element subsets of A. S has |A|^2-|A| elements. |A|^2 > |G|^2/4-|G|/2. I want to show that at least |G| of the products of the of the two-element sets in A are distinct. This approach does not seem like it will work...

Maybe I should consider cases. If the e is in A, then any element a in A equal a*e. If e is not in A, I'm not really sure what to do...

 

[morphism]

Here's a hint: inverses are key.

 

[ehrenfest]

Here's a hint: inverses are key.

Very nice hint. Short but helpful.

Let g be any element of G. Then the left coset g*A^{-1} has order greater than |G|/2, so it must intersect A, since A also has the same order. That means there must be elements in A, a_1 and a_2, such that g*a_1^{-1}=a_2. That is g=a_1*a_2.

Is that right?

 

[morphism]

Looks good to me (although I believe the word "coset" is reserved for subgroups)!