Lie powers of relation modules for groups

Abstract

Motivated by applications to abstract group theory, we study Lie powers of relation modules. The relation module associated to a free presentation G = F /N of a group G is the abelianization N ab = N/ [N, N] of N, with G-action given by conjugation in F . The degree n Lie power is the homogeneous component of degree n in the free Lie ring on N ab (equivalently, it is the relevant quotient of the lower central series of N). We show that after reduction modulo a prime p this becomes a projective G-module, provided n > 1 and n is not divisible by p.Crown

Free central extensions of groups and modular Lie powers of relation modules

“…In the special case where N = F this gives the following. This extends a recent result on projectivity of modular Lie powers of relation modules: For an arbitrary positive integer n ≥ 2, the Lie power L n (M p ) is a projective (Z/pZ)G-module for all primes p that do not divide n [8,Corollary]. Notice that this result holds without any restriction on the group G. The paper [8] was written with applications to (1.1) in mind.…”

“…This extends a recent result on projectivity of modular Lie powers of relation modules: For an arbitrary positive integer n ≥ 2, the Lie power L n (M p ) is a projective (Z/pZ)G-module for all primes p that do not divide n [8,Corollary]. Notice that this result holds without any restriction on the group G. The paper [8] was written with applications to (1.1) in mind. Combining the result of [8] with Theorem 2 yields the following corollary, which is exactly what is required to deduce Theorem 1.…”

“…Notice that this result holds without any restriction on the group G. The paper [8] was written with applications to (1.1) in mind. Combining the result of [8] with Theorem 2 yields the following corollary, which is exactly what is required to deduce Theorem 1.…”

“…In Section 2 we deduce Theorem 1 from Corollary 2, and the remaining sections are devoted to the proof of Theorem 2. For the latter, we expand the results of [8] to a point that makes it possible to invoke a deep result on modular Lie powers, the Bryant-Schocker decomposition, a special case of which was also the key ingredient in [7]. …”

RELATIVE RELATION MODULES OF FINITE ELEMENTARY ABELIAN p-GROUPS

“…Gaschutz [1], Gruenberg [2], Kovacs and Stohr [4], Mittal and Passi [5], and others have studied relation modules. Relative relation modules have been studied by Kimmerle [3], Yamin [8,9], and Sharma and Yamin [7].…”