Normalizers

Normalizers are terms used in several disciplines to denote objects or procedures that bring entities into a standard or compatible form. In mathematics, a normalizer of a subgroup H inside a group G is the set N_G(H) = {g in G | gHg^{-1} = H}. It is a subgroup of G that contains H, and within N_G(H) the subgroup H is normal. The normalizer captures how far H can be rotated by conjugation without leaving its own structure.

Key properties include that H is normal in N_G(H), and N_G(H) is the largest subgroup of G in which H is normal. The conjugates of H in G are in one-to-one correspondence with the cosets of N_G(H). By contrast, the centralizer C_G(H) = {g in G | gh = hg for all h in H} is the largest subgroup that commutes elementwise with H, and is contained in the normalizer.

In data processing and machine learning, normalizers refer to methods that scale features to a common range or distribution. Common techniques include min-max normalization, which rescales features to a fixed interval such as [0, 1], and standardization (z-score normalization), which centers data to mean zero and unit variance. Normalization can improve numerical stability and convergence of algorithms but must be applied consistently to train, validation, and test data to avoid leakage.

In probability and statistics, a normalizing constant (or normalizer) is a factor that adjusts a function to integrate to one, turning it into a probability density or mass function. The term appears in Bayesian inference and in models that require proper probability distributions.