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On the Fixed Points of the Lie Ring Associated with a Free Presentation of a Finite Group
Author(s) -
Zerck Rainer
Publication year - 1989
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/21.4.381
Subject(s) - subring , mathematics , ring (chemistry) , free group , generating set of a group , combinatorics , rank (graph theory) , invariant (physics) , group ring , group (periodic table) , pure mathematics , discrete mathematics , chemistry , mathematical physics , geometry , organic chemistry
Let l → R → F → G → l be a presentation of the finite non‐trivial group G with finitely generated free group F . Then G acts by conjugation on the Lie ring r = ⊕ c ⩽ 1 γ c R /γ c +1 R (γ c R is the c th term of the lower central series of R ), which is free of finite rank by the classical result of Magnus and Witt. In this note the Lie subring of r consisting of all G ‐invariant elements is shown to be free with infinite free generating set.