Do centralizer and normalizer require H to be a subgroup?

[loly2]

in the definition of centalizer of subset H , Is H should be a non-empty subset?
also, Is H should be subgroup in the normalizer of H?

 

[HallsofIvy]

There is always at least one member of the group G that commutes with every member! What is it? The centralizer of a subgroup it always a subgroup itself