Proof that S is a generating set.

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Homework Help Overview

The discussion revolves around proving that a subset S of a finite group G, with an order greater than half the order of G, is a generating set for G. The context is within the subject area of Algebra, specifically group theory.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the properties of the subgroup H generated by S and explore the implications of its order relative to G. Questions arise about the disjoint nature of sets and the relationship between H and cosets.

Discussion Status

The discussion is active, with participants sharing insights about the properties of groups and subgroups. Some guidance has been offered regarding the relationship between H and the element x not in H, and the implications of their orders. There is an ongoing exploration of the proof's logic without a definitive consensus yet.

Contextual Notes

Participants note the absence of knowledge regarding cosets and Lagrange's theorem, which may influence their approaches to the proof. There is an emphasis on proving the statement from foundational principles.

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Homework Statement


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.

Homework Equations


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

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.
 
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kasperrepsak said:

Homework Statement


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.

Homework Equations


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

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.

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:
Dick said:
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 that's not in H. Can you show H and xH are disjoint sets? Sure, xH is a coset. But you don't have to know that to do the proof.

Thanks for the fast reply ! : ) Thank you I will work with that.
 
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?
 
kasperrepsak said:
Ok so I know that H and xH must be disjoined sets (easy to proof). H and xH have the same number of elements. They both must be in G. But since the order of H is greater than half the order of G, H and xH combined would have more elements than G which isn't possible. Therefore H must be the whole group G. Is this right?

Absolutely correct.
 
Ok thanks again for your help : ).
 

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