Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-27T18:32:57.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Focal Series in Finite Groups

Published online by Cambridge University Press:  20 November 2018

D. G. Higman
D. G. Higman*
Affiliation:
McGill University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If there is given a subgroup 5 of a (finite) group G, we may ask what information is to be obtained about the structure of G from a knowledge of the location of S in G. Thus, for example, famed theorems of Frobenius and Burnside give criteria for the existence of a normal subgroup N of G such that G = NS and 1 = N ⋂ S, and hence in particular for the non-simplicity of G. To aid in locating S in G, and to facilitate exploitation of the transfer, we single out a descending chain of normal subgroups of S. Namely, we introduce the focal series of S in G by means of the recursive formulae

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 477 - 497
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Baer, R., Representations of groups as quotient groups I, Trans. Amer. Math. Soc, 58 (1945), 295–347.Google Scholar
2. Baer, R., Group elements of prime power index, Trans. Amer. Math. Soc, 75 (1953), 20–47.Google Scholar
3. Frobenius, G., Über auflosbare Gruppen IV, S. B. preuss. Akad. Wiss. (1901), 1216–1230.Google Scholar
4. Grim, O., Beitrdge zur Gruppentheorie I, J. reine angew. Math., 174 (1935), 1–14.Google Scholar
5. Grim, O., Beitrdge zur Gruppentheorie II, J. reine angew. Math., 186 (1945), 165–169.Google Scholar
6. Grim, O., Beitrdge zur Gruppentheorie III, Math. Nachr., 1 (1948), 1–24.Google Scholar
7. Hall, P., A characteristic property of finite solvable groups, J. London Math. Soc, 12 (1937), 198–200.Google Scholar
8. Schur, I., Neuer Beweis eines Satzes Über endliche Gruppen, S. B. preuss. Akad. Wiss. (1902), 1013–1019.Google Scholar
9. Zassenhaus, H., The theory of groups (New York, 1949).Google Scholar