Let G be a group and H a subgroup. Prove if [G:H]=2, then H is normal.

In summary, the conversation discusses proving that if the index of H in G is equal to 2, then H is a normal subgroup of G. The key points mentioned are that the index of H in G means there are only two left and right cosets of H, and that G\H represents the set-theoretic difference between G and H. This information helps in understanding the proof.
  • #1
mathmajor2013
26
0
Let G be a group and H be a subgroup of G. Prove that if [G:H]=2, then H is a normal subgroup of G.
 
Physics news on Phys.org
  • #2
The same as in my other reply:

1) this should belong in the homework forums
2) what did you try?
 
  • #3
I'm lost on this one. It doesn't make sense how the number of left cosets corresponds to the normality. #gH=#Hg doesn't seem like it necessarily means that gH=Hg.
 
  • #4
That [G:H]=2 means that there are only two left cosets of H. Also, it means that there are only two right cosets of H: H and G\H.

Thus gH is H or G\H, and for Hg thesame thing. Does this help you?
 
  • #5
I thought that G/H was the set of left or right cosets, not a coset itself? But yes that does help, thank you!
 
  • #6
No, I mean G\H, not G/H. With G\H, I mean the set-theoretic difference, i.e. everything in G which is not in H.
 
  • #7
Oh I see. Yes! Thank you.
 
  • #8
(Thread moved and OP pinged)
 

1. What does [G:H]=2 mean in the context of a group and subgroup?

In group theory, [G:H] represents the index of a subgroup H in a group G. It is the number of cosets of H in G. When [G:H]=2, it means that there are only two distinct cosets of H in G.

2. Can you explain the concept of a normal subgroup?

A normal subgroup is a subgroup that is invariant under conjugation by elements of the larger group. This means that for any element g in the group G and any element h in the subgroup H, the element ghg-1 is also in H. In other words, the left and right cosets of a normal subgroup are equal.

3. How does the proof of [G:H]=2 imply that H is normal?

Since [G:H]=2, there are only two distinct cosets of H in G. This means that the left and right cosets are equal, and therefore, H is invariant under conjugation by elements of G. Therefore, H is a normal subgroup.

4. Is [G:H]=2 a sufficient condition for H to be normal in G?

Yes, if [G:H]=2, then H is guaranteed to be a normal subgroup. However, it is not a necessary condition. There are cases where a subgroup may be normal even if [G:H] is not equal to 2.

5. Can you provide an example of a group and subgroup where [G:H]=2 and H is not normal?

Yes, consider the group G = {1, 2, 3, 4, 5, 6} under addition modulo 6. Let H = {1, 3}. The index of H in G is [G:H]=2, but H is not normal in G since 2H = {2, 4} and 2H ≠ H2 = {3, 5}.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
988
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
764
  • Calculus and Beyond Homework Help
Replies
7
Views
888
  • Calculus and Beyond Homework Help
Replies
1
Views
885
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
Back
Top