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

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A subgroup H is defined as normal if gHg^(-1) = H for all elements g in the group. If H is the unique subgroup of a certain order n, it must be normal because all conjugates xHx^(-1) will also have order n. The discussion clarifies that a subgroup is normal if its left and right cosets are identical. This property allows for the definition of a group operation on the cosets, forming a new group G/H. Understanding these concepts is essential for grasping the structure of groups and their subgroups.
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Normal subgroups??

Normal subgroups?
 
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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.
 
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.
 
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.
 

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