Nonnormal

Nonnormal is an adjective used in mathematics and the sciences to designate that an object does not possess a property designated by the word normal. The term is common in contexts that require a precise standard, such as normal subgroups in group theory, normal operators in functional analysis, or normal probability distributions in statistics.

In group theory, a subgroup H of a group G is normal if it is invariant under conjugation by every element of G; equivalently, gHg^{-1} = H for all g in G. A nonnormal subgroup is a subgroup that fails this condition. For example, in the symmetric group S3, the subgroups generated by a transposition are not normal, while the subgroup A3 of even permutations is normal.

In functional analysis, a normal operator A satisfies AA* = A*A. Operators that do not satisfy this equality are called nonnormal. Nonnormal operators can exhibit spectral and dynamical behavior that differs from normal operators; for instance, the 2x2 matrix [[0,1],[0,0]] is not normal.

In statistics, a nonnormal distribution refers to any distribution that does not follow the normal (Gaussian) distribution. Nonnormal data may be skewed or heavy-tailed, which can affect the validity of parametric tests that assume normality. Analysts may apply data transformations or use nonparametric methods.

Outside these fields, nonnormal is used as a general descriptor for systems, models, or data that do not conform to an established "normal" form. The term is not a formal property in all disciplines, and its meaning is context dependent.

---