Is H a Subgroup of N if |H| and |G:N| are Relatively Prime?

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Discussion Overview

The discussion revolves around the question of whether a subgroup H of a finite group G is contained within a normal subgroup N of G, given that the orders of H and the quotient group G:N are relatively prime. Participants explore various approaches to demonstrate this relationship, focusing on group properties and orders.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant suggests using the fact that since N is normal, the product HN is a subgroup of G.
  • Another participant proposes examining the element h in H and its relation to the quotient group G/N, specifically questioning the order of hN in G/N.
  • A later reply discusses the order of hN, noting that it divides the order of G/N, but expresses uncertainty about whether it also divides |H|.
  • Further clarification is provided regarding the order of h and its implications for the intersection of H and N.
  • One participant expresses gratitude for the assistance received, indicating a breakthrough in understanding.

Areas of Agreement / Disagreement

The discussion appears to involve multiple viewpoints and approaches, with no consensus reached on the final conclusion regarding the subgroup relationship.

Contextual Notes

Participants express uncertainty about the divisibility of certain orders and the implications of the normal subgroup property, indicating potential limitations in their reasoning.

Who May Find This Useful

This discussion may be useful for students studying group theory, particularly those interested in subgroup properties and the relationships between group orders.

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Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Show that if |H| and |G:N| are relatively prime then H is a subgroup of N.

I have tried using the fact that since N is normal, HN is a subgroup of G.
Suposing that H is not contained in N, I tried finding a common factor for |H| and |G:N|.
Numbers that divide |H| are |HnN|, |H:HnN| and |H|.
Numbers that divide |G:N| are |G:HN| and |HN:N|.
I'm stuck.

I would appreciate any suggestions.
Thanks
 
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Take a h\in H. You want to show that h\in N. Now, what can you say about h+N\in G/N?? What is its order??
 
The order of hN in G/N is the smallest integer k such that h^k = n for some n in N.
We know that k divides |G/N|. I don't know if k divides |H|. We know that h^k is in the intersection of H and N.
 
What I meant was: let k be the order of h. Then we know that h^k=e. And also (hN)^k=N. So the order of hN divides k. What can you conclude?
 
Thank you micromass,
I got it.
 

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