Efficient characteristic refinements for finite groups

Abstract

Filters were introduced by J.B. Wilson in 2013 to generalize work of Lazard with associated graded Lie rings. It holds promise in improving isomorphism tests, but the formulas introduced then were impractical for computation. Here, we provide an efficient algorithm for these formulas, and we demonstrate their usefulness on several examples of p-groups.

Compatible filters with isomorphism testing

“…Example 7.2. We consider a group examined in [ELGO,Section 12.1] and [M1,Section 5]. For a fixed odd prime p, we define a p-group G by a power-commutator presentation, where all trivial commutators are omitted G = g 1 , ..., g 13 | [g 10 , g 6 ] = g 11 , [g 10 , g 7 ] = g 12 , [g 2 , g 1 ] = [g 4 , g 3 ] = [g 6 , g 5 ] = [g 8 , g 7 ] = [g 10 , g 9 ] = g 13 , exponent p .…”

“…While the applications vary, a common theme is to make group-theoretic problems easier by employing linear algebra in the context of the Lie ring. In particular, this helps in the study of isomorphism and automorphism problems for groups [ELGO,H1,K1,M1,M3]. A recent development, described in [W], generalizes approaches from Magnus [M5, M6] and Lazard [L].…”

A spectral theory for transverse tensor operators

“…Here, effort is placed on derived algebras like adjoint rings with involutions, Lie and Jordan algebras, and groups of isometries. This has enabled projects ranging across ωstability and finite Morley rank in algebra [M3, M4], group isomorphism [BW14, BMW17, IQ15, LW12, LQ17, Wil], intersecting classical groups [BW12, GG07, BF87], Krull-Schmidt type theorems [Wil09,Wil12], obstructing existence of characteristic subgroups [Mag17,Mag15,GPS11], and as models of Hermitian categories [BFFM14,Wil13]. While these approaches have been fruitful, the techniques are all specialized by design and mostly concern 3-tensors and bilinear maps.…”

Testing isomorphism of graded algebras

“…This general treatment allows our algorithms to take as input more refined gradings on the given algebras that often decompose them into smaller pieces. The development of such refined gradings is an emerging area of study that may lead to faster isomorphism tests; see [16], for example.…”