Hi Stormey,
Suppose G is abelian and H is any subset of G. Isn't it true that ghg^{-1} is in H for all g in G and all h in H?
Perhaps your teacher wants you to prove that <H>, the subgroup generated by H, is a normal subgroup.
Suppose is a group, and is some subset of .
Is it true that if and : , then is a normal subgroup?
Or that has to be a subgroup (not just any subset) for this statment to be true?
I'm asking because I need to prove it, and I not 100 percent sure my teacher didn't make a mistake here...
Thanks in advance.
Hi Stormey,
Suppose G is abelian and H is any subset of G. Isn't it true that ghg^{-1} is in H for all g in G and all h in H?
Perhaps your teacher wants you to prove that <H>, the subgroup generated by H, is a normal subgroup.
Stormey,
One counterexample is always enough to refute a statement. However, your statement is also false for non-abelian groups.
Example: Let G=S_{3} be the symmetric group on {1,2,3}, x the 3 cycle (123) and H = {x,x^{-1}}. I leave it to you to prove that ghg^{-1} is in H for all g in G and h in H. "Clearly", H is not a subgroup of G.