# Homework Help: Proof that S is a generating set.

1. Feb 24, 2013

### kasperrepsak

1. The problem statement, all variables and given/known data
I have recently started a new course in Algebra. I have to proof that if S is a subset of a finite group G, with an order greater than half the order of G, S is a generating subset for G.

2. Relevant equations
I haven't had cosets nor Lagrange theorem so I suppose I should try to prove it from scratch.

3. The attempt at a solution
I have put my mind to the problem for a while now but didnt come up with any meaningful clues. I would be very thankful for any tips.

2. Feb 24, 2013

### Dick

Let H be the group generated by S. That's a subgroup of G and it has at least as many elements as S. Suppose H isn't the whole group G. Then there is an element x of G that's not in H. Can you show H and xH are disjoint sets and they have the same number of elements? Sure, xH is a coset. But you don't have to know that to do the proof.

Last edited: Feb 24, 2013
3. Feb 24, 2013

### kasperrepsak

Thanks for the fast reply ! : ) Thank you I will work with that.

4. Feb 24, 2013

### kasperrepsak

Ok so I know that H and xH must be disjoined sets (easy to proof), they have the same number of elements and both must be in G. But since the order of H is greater than half the order of G, H and xH unified would have more elements than G which is a contradiction. Therefore H must be the whole group G. Is this right?

5. Feb 24, 2013

### Dick

Absolutely correct.

6. Feb 24, 2013

### kasperrepsak

Ok thanks again for your help : ).