Normalizer

Problem Definition

Given two groups $S$ and $G$, the normalizer of $S$ in the group $G$ is defined as:

$\mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}.$

Finding the normaliser of a group is a difficult problem in computational group theory. A commonly considered special case is where $G$ is the symmetric group, even here the problem is still difficult.

Attempts to solve the normaliser problem have traditionally taken two routes, either producing more practically efficient algorithms for the general case, or considering specialised theory and algorithms for special cases of $S$.

Brief overviews of algorithms for the normalizer can be found in (Holt et al., 2005) and (Seress, 2003)

Algorithms

Improved theory and algorithms for finding the normalizer of primitive groups can be found in:

(Siccha, 2020) Towards Efficient Normalizers of Primitive Groups. repository.

Groups

Finite Non-Abelian Simple Groups

Primitive Groups

Primitive Affine Groups

Primitive of Type PA Groups

References

1. Holt, D. F., Eick, B., & O’Brien, E. A. (2005). Handbook of computational group theory (p. xvi+514). Chapman & Hall/CRC, Boca Raton, FL. https://doi.org/10.1201/9781420035216
2. Seress, Á. (2003). Permutation group algorithms (Vol. 152, p. x+264). Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511546549
3. Siccha, S. (2020). Towards Efficient Normalizers of Primitive Groups.

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