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On Groups with all Composition Factors Isomorphic

Published online by Cambridge University Press:  20 November 2018

R. Bercov
R. Bercov*
Affiliation:
University of Alberta, Edmonton
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By the celebrated theorem of Jordan [3] and Hölder [2], there is associated with each finite group G a family of distinct simple groups Hi. such that every composition series of G has ni factor groups isomorphic to Hi and no others. We denote the collection of pairs (Hi, ni) by CF(G). Conversely, given k pairs (Hi, ni), we may construct by an easy direct product procedure a group G with CF(G) = { (Hi, ni) | i =1,…, k}. The composition factors, of course, do not in general determine the group.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 9 , Issue 4 , October 1966 , pp. 413 - 415
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Hall, Marshall Jr., The Theory of Groups. MacMillan, (1959).Google Scholar
2. Hölder, O., Zuruckfuhrung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen. Math. Ann. 34, (1889), pages 26-56.Google Scholar
3. Jordan, C., Commentaire sur Galois. Math. Ann. 1, (1869), pages 141-160.Google Scholar
4. Schreier, O. and van der Waerden, B.L., Die automorphismen der projektiven Gruppen. Abh. Math. Sem. Hamburg 6, (1928), pages 303-322.Google Scholar
5. Scott, W.R., Group Theory. Prentice-Hall, (1964).Google Scholar