Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-17T07:16:48.917Z Has data issue: false hasContentIssue false
  • English
  • Français

On a property of verbal and marginal subgroups

Published online by Cambridge University Press:  24 October 2008

P. W. Stroud
P. W. Stroud
Affiliation:
Trinity College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

Let G be a group, and let φ (x1, …, xn) be any word in the variables x1, …, xn. We denote by φ(G) and φ*(G) respectively the corresponding verbal and marginal subgroups of G, which are defined as follows.

(i) φ(G)is the subgroup generated by all elements of the form φ(a1; …, an) with ai in G(for i = 1, 2, …, n);

(ii) an element b of G lies in φ*(G) if and only if

for all choices of a1, …, an in G and i= 1, 2, …,n.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 1 , January 1965 , pp. 41 - 48
Copyright
Copyright © Cambridge Philosophical Society 1965

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Baer, R.Endlichkeitskriterien für Kommutatorgruppen. Math. Ann. 124 (1952), 161–177.Google Scholar
(2)Hall, P.Finite-by-nilpotent groups. Proc. Cambridge Philos. Soc. 52 (1956), 611–616.Google Scholar
(3)Hall, P.Finiteness conditions for soluble groups. Proc. London Math. Soc. (3) 4 (1954), 419–436.Google Scholar
(4)Hall, P.On the finiteness of certain soluble groups. Proc. London Math. Soc. (3) 9 (1959), 595–622.CrossRefGoogle Scholar
(5)Schur, I.Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitionen. J. Reine Angew. Math. 127 (1904), 20–50.Google Scholar
(6)Smirnov, D. M.On groups of automorphisms of soluble groups. Mat. Sb. 32 (74) (1953), 365–384 (in Russian).Google Scholar
(7)Weichsel, P. M.On critica p-groups. Proc. London Math. Soc. (3) 14 (1964), 83–100.Google Scholar
(8)Zassenhaus, H.The theory of groups (Chelsea; New York, 1949).Google Scholar