- #1
Oster
- 85
- 0
I have to prove that every subgroup H of a group G with index(number of distinct cosets of the subgroup) 2 is normal.
I don't know how to start :'(
Please help.
I don't know how to start :'(
Please help.
In summary, to prove that every subgroup H of a group G with index 2 is normal, you must show that gN=Ng for all elements g. If g comes from H, both gH and Hg are equal to H. If g comes from the complement of H, both gH and Hg are equal to the complement of H.
- #2
- 22,183
- 3,321
Hi Oster! 
You must prove that gN=Ng for all g. But what exactly are the cosets of N??
You must prove that gN=Ng for all g. But what exactly are the cosets of N??
- #3
Oster
- 85
- 0
If the element g comes from H then both gH and Hg are equal to H.
If g comes from H complement then i know it must represent the "other" coset. Since cosets partition G, i know both these cosets must be equal to H complement.
More or less correct?
If g comes from H complement then i know it must represent the "other" coset. Since cosets partition G, i know both these cosets must be equal to H complement.
More or less correct?
- #4
- 22,183
- 3,321
Yes, that's good!
1. What does it mean for a subgroup to have an index of 2?
For a given group, the index of a subgroup is the number of cosets (distinct left or right translates) of the subgroup in the group. If a subgroup has an index of 2, it means that there are exactly two cosets of the subgroup in the group.
2. Why is it important for a subgroup to have an index of 2?
A subgroup having an index of 2 indicates that the subgroup is a normal subgroup, which means it is invariant under conjugation by elements of the larger group. This allows for simpler and more efficient computations and can provide insights into the structure of the group.
3. How is the normality of a subgroup related to its index?
The statement "every subgroup of index 2 is normal" means that for any group, if a subgroup has an index of 2, then it is guaranteed to be a normal subgroup. In other words, the index of a subgroup is a sufficient condition for it to be normal.
4. Can a subgroup have an index of 2 and not be normal?
No, a subgroup cannot have an index of 2 and not be normal. This is because the index of a subgroup being 2 is a necessary and sufficient condition for it to be normal. If a subgroup has an index of 2, then it must be normal.
5. How does the statement "every subgroup of index 2 is normal" apply to specific groups?
This statement applies to all groups, regardless of their specific structure or properties. It is a fundamental result in group theory that is applicable to any group with subgroups of index 2.
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