Are Normal Subgroups Defined by Equality of Left and Right Cosets?

[raj123]

Normal subgroups??

Normal subgroups?

 

[matt grime]

H is normal if gHg^(-1)=H for all g. If H is a subgroup of some order, then so is gHg^(-1). End of hint.

 

[brownnrl]

Ah so

If H is unique subgroup of order n (no others) it must be normal as all other xHx^(-1) must be of that same order n.

I was thrown by the 10 or 20 in the problem, but it could really be any order n.

Thank you very much for the hint. I saw the disclaimer after I posted about the homework, so I'm sorry if this question wasn't up to par.

 

[HallsofIvy]

Also, a subgroup, H, of a group, G, is a normal subgroup if and only if the "left cosets" and "right cosets" are the same. A result of that is that we can define the group operation on the cosets (if p is in coset A and q is in coset B then AB is the coset that contains pq) in such away that the collection of cosets is a group in its own right: G/H.