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Details for article 1 of 13 found articles
| Title: |
A GENERALIZATION OF COMPLETE GROUPS |
| Author: |
Gibson, Gregory A. |
| Appeared in: |
Communications in algebra |
| Paging: | Volume 24 (1996) nr. 8 pages 2501-2514 |
| Year: |
1996 |
| Contents: |
Let G be a finite group and let G be the semi-direct product of a normal subgroup N and a subgroup K. In [1], conditions were found which are equivalent to the existence of a normal complement to N in G. We consider the structure of groups N for which the above condition always holds. Thus we use Bechtell's results to gain information on groups N such that if G is a semi-direct product of N and a subgroup K, then N is a direct factor of G, for all G. It is an old result that a group N is complete if and only if whenever N is a normal subgroup of G, then N is a direct factor of G, [4]. Hence it is not surprising that complete groups are part of our result. Moreover a group N is complete if and only if N is isomorphic to Aut(N) under the mapping σ(n) = σn, where σn is the inner automorphism induced by n. This remark leads us to consider groups N which contain a subgroup H such that H is isomorphic to Aut(N) under σ: H → Aut(N). All groups considered here are finite. The results found here do not parallel the results found in the author's dissertation for Lie algebras. There it is shown that only complete Lie algebras have the desired property. Thus, these results provide an example of when the theory of Lie algebras diverges from that of groups. |
| Publisher: |
Taylor & Francis |
| Source file: |
Elektronische Wetenschappelijke Tijdschriften |
Details for article 1 of 13 found articles
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