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

  • Thread starter symbol0
  • Start date
  • Tags
    Subgroup
In summary, if G is a finite group, H is a subgroup of G, and N is a normal subgroup of G, then if the orders of H and G/N are relatively prime, H is a subgroup of N. This can be proven by considering the order of hN in the quotient group G/N and using the fact that it must divide the order of h in H.
  • #1
symbol0
77
0
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
 
Physics news on Phys.org
  • #2
Take a [itex]h\in H[/itex]. You want to show that [itex]h\in N[/itex]. Now, what can you say about [itex]h+N\in G/N[/itex]?? What is its order??
 
  • #3
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.
 
  • #4
What I meant was: let k be the order of h. Then we know that [itex]h^k=e[/itex]. And also [itex](hN)^k=N[/itex]. So the order of hN divides k. What can you conclude?
 
  • #5
Thank you micromass,
I got it.
 

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

What is a subgroup?

A subgroup is a subset of a group that still satisfies the four defining properties of a group: closure, associativity, identity element, and inverse element.

How do you prove that H is a subgroup of N?

To prove that H is a subgroup of N, you must show that H satisfies the three subgroup criteria: closure, associativity, and inverse elements. This can be done by showing that for any two elements in H, their product and inverse are also in H.

What is the notation for denoting a subgroup?

The notation for denoting a subgroup is H⊂N, where H is the subgroup and N is the original group. This means that H is a subset of N and satisfies the three subgroup criteria.

Can a subgroup have a different identity element than the original group?

No, a subgroup must have the same identity element as the original group. If a subgroup has a different identity element, it cannot be a subgroup as it would break the subgroup criteria of having closure and associativity.

What is the significance of proving that H is a subgroup of N?

Proving that H is a subgroup of N allows us to better understand the structure and properties of the original group N. It also helps us to identify patterns and relationships within the group, which can be useful in solving mathematical problems.

Similar threads

MHB Why Can't a Proper Normal Subgroup Contain a Sylow Normalizer in a Group?
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Show that the groups are abelian
    • Linear and Abstract Algebra
Replies
1
Views
970
MHB Can You Prove These Properties of Normal Subgroups in Group Theory?
    • Linear and Abstract Algebra
Replies
2
Views
1K
I Is G simple?Is G simple? A different approach using Sylow theorems
    • Linear and Abstract Algebra
Replies
5
Views
2K
MHB Proving N(H) is a Subgroup of G Containing H
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Trying to get the point of some Group Theory Lemmas
    • Linear and Abstract Algebra
Replies
3
Views
992
MHB 4 part problem regarding homomorphisms
    • Linear and Abstract Algebra
Replies
1
Views
1K
I What is the meaning of N(H) in subgroup notation?
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Normalizer of a subgroup of prime index
    • Linear and Abstract Algebra
Replies
6
Views
2K
MHB Proof: Every Subgroup of Cyclic $H$ is Normal in $G$
    • Linear and Abstract Algebra
Replies
12
Views
3K

Hot Threads

Recent Insights

Back
Top