caa a22 4500 386385408 CHVBK 20180307112056.0 cr unu---uuuuu 161130e198907 xx s 000 0 eng 10.1017/S0305004100067955 doi S0305004100067955 pii (NATIONALLICENCE)cambridge-10.1017/S0305004100067955 Stöhr Ralph Akademie der Wissenschaften der DDR, Karl- Weierstraβ-Institut für Mathematik, Mohrenstraβe 39, Berlin, DDR-1086, German Democratic Republic On elements of order four in certain free central extensions of groups [Elektronische Daten] [Ralph Stöhr] Let F be a non-cyclic free group, R a normal subgroup of F and G = F/R, i.e. where π is the natural projection of F onto G, is a free presentation of G. Let R′ denote the commutator subgroup of R. The quotient F/[R′,F] is a free central extension of the group F/R′, the latter being a free abelianized extension of G. While F/R′ is torsion-free (see, e.g. [2], p. 23), elements of finite order may occur in R′/[R′,F], the kernel of the free central extension (l·2). Since C. K. Gupta [1] discovered elements of order 2 in the free centre-by-metabelian group F/[F″,F] (i.e. (1·2) in the case R = F′), torsion in F/[R′,F] has been studied by a number of authors (see, e.g. [4-13]). Clearly the elements of finite order in F/[R′,F] form a subgroup T of the abelian group R′/[R′,F]. It will be convenient to write T additively. By a result of Kuz'min [5], any element of T has order 2 or 4. Moreover, it was pointed out in [5] that elements of order 4 may really occur. On the other hand, it has been shown in [11] that, if G has no 2-torsion, then T is an elementary abelian 2-group isomorphic to H4(G, ℤ2). So if T contains an element of order 4, then G must have 2-torsion. We also mention a result of Zerck [13], who proved that 2T is an invariant of G, i.e. it does not depend on the particular choice of the free presentation (1·1). Copyright © Cambridge Philosophical Society 1989 Mathematical Proceedings of the Cambridge Philosophical Society Cambridge University Press 106/1(1989-07), 13-28 0305-0041 106:1<13 1989 106 PSP https://doi.org/10.1017/S0305004100067955 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0305004100067955 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Stöhr Ralph Akademie der Wissenschaften der DDR, Karl- Weierstraβ-Institut für Mathematik, Mohrenstraβe 39, Berlin, DDR-1086, German Democratic Republic NATIONALLICENCE 773 0- Mathematical Proceedings of the Cambridge Philosophical Society Cambridge University Press 106/1(1989-07), 13-28 0305-0041 106:1<13 1989 106 PSP CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge