A fast isomorphism test for groups whose Lie algebra has genus 2

Abstract

Motivated by the need for efficient isomorphism tests for finite groups, we present a polynomial-time method for deciding isomorphism within a class of groups that is well-suited to studying local properties of general finite groups. We also report on the performance of an implementation of the algorithm in the computer algebra system magma.1991 Mathematics Subject Classification. 20D15, 20D45, 15A22, 20B40.

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“…The algorithms of [21] have a complexity of O(p + e 2ω log 2 p), where 2 ≤ ω < 3 is the exponent of matrix multiplication, see [31,Chapter 12]. In the special case of groups in H 2 p,e , Brooksbank, Maglione, and the author give an O(p 3 + e ω log 2 p)-time algorithm [7]. Note that it takes Ω(e 2 log 2 p) bits to input the groups in H 2 p,e by any of the standard methods, including matrices, presentations, or permutations.…”

“…These algorithms have been implemented in the computer algebra system Magma [5] and decide isomorphism for groups as large as n = 5 256 in about an hour on a general-purpose personal computer [7, Figure 1.1]. The algorithms in [7] further give a complete invariant for the groups in H 2 p,e , which is a homogeneous polynomial of degree e in F p [x, y]. This implies that the number of isomorphism classes in H 2 p,e is at most p e ; see [7,Proposition 9.2].…”

“…The algorithms in [7] further give a complete invariant for the groups in H 2 p,e , which is a homogeneous polynomial of degree e in F p [x, y]. This implies that the number of isomorphism classes in H 2 p,e is at most p e ; see [7,Proposition 9.2]. Some of the steps in the proof of Theorem 1.5 can be extracted as special cases of results in [21,8,7].…”

“…This implies that the number of isomorphism classes in H 2 p,e is at most p e ; see [7,Proposition 9.2]. Some of the steps in the proof of Theorem 1.5 can be extracted as special cases of results in [21,8,7]. However, there is an increased need to provide an introduction into the methods in those works.…”

“…Z(G) denotes the center of G. We write G p = x p | x ∈ G . As in [7], we borrow terminology from Lie theory and say the genus of a group G of nilpotence class 2 and exponent p is…”