Pronormal subgroup
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In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, (Doerk & Hawkes 1992, I.§6).
A subgroup is pronormal if each of its conjugates is conjugate to it already in the subgroup generated by it and its conjugate. That is, H is pronormal in G if for every g in G, there is some k in the subgroup generated by H and Hg such that Hk = Hg.
Here are some relations with other subgroup properties:
- Every normal subgroup is pronormal.
- Every Sylow subgroup is pronormal.
- Every pronormal subnormal subgroup is normal.
- Every abnormal subgroup is pronormal.
- Every pronormal subgroup is weakly pronormal, that is, it has the Frattini property
- Every pronormal subgroup is paranormal, and hence polynormal
[edit] References
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This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (September 2008) |
- Doerk, Klaus; Hawkes, Trevor (1992), Finite soluble groups, de Gruyter Expositions in Mathematics, 4, Berlin: Walter de Gruyter & Co., MR1169099, ISBN 978-3-11-012892-5
