normalizer
Given a subgroup sitting inside a bigger group, you might want to enlarge the surroundings until the subgroup becomes normal — invariant under conjugation by everything around it. The normalizer is the largest such surrounding: the biggest subgroup in which your chosen subgroup is normal.
Formally, for a subgroup H of G, the normalizer is N_G(H) = { g in G : g H g^{-1} = H }. It is a subgroup of G containing H, and H is normal in N_G(H) by construction; moreover N_G(H) is the largest subgroup of G with this property. Note the difference from the centralizer: the normalizer only asks that conjugation send the set H back to itself, not that it fix each element individually, so C_G(H) is always contained in N_G(H).
Normalizers organize how a subgroup interacts with the rest of the group. The number of conjugates of H equals [G : N_G(H)] by orbit-stabilizer applied to the conjugation action on subgroups, so a self-normalizing subgroup (N_G(H) = H) has the maximum possible number of conjugates. Normalizers of Sylow subgroups, in particular, are central to Sylow theory and to counting arguments throughout finite group theory.
In S_3, take H = { e, (1 2) }. Its normalizer is H itself, since conjugating (1 2) by a 3-cycle yields a different transposition, moving H to a different subgroup. So H has [S_3 : H] = 3 conjugate subgroups, the three order-2 subgroups generated by the transpositions.
A self-normalizing order-2 subgroup of S_3.