In my text's set of definitions we have

The centralizer of a subset A of a group G: $\displaystyle C_G(A) = \{ g \in G|gag^{-1} = g ~\forall a \in A \}$

The center of a group G: $\displaystyle Z(G) = \{ g \in G | gz = zg ~ \forall z \in G \}$

The normalizer of a subset A of a group G: $\displaystyle N_G (A) = \{g \in G|g A g^{-1} = A \}$

Now the question:Whydo these groups have these names? Is there some intrinsic meaning to them or are they just terms that have become accepted?

Thanks!

-Dan