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Abnormal subgroup

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A subgroup of a group such that for any element . Here is the subgroup generated by and its conjugate subgroup . As an example of an abnormal subgroup of a finite group one can take the normalizer (cf. Normalizer of a subset) of any Sylow -subgroup , and even any maximal subgroup which is not normal in . In the theory of finite solvable groups (cf. Solvable group), where many important classes of subgroups are abnormal, use is made of the concept of a subabnormal subgroup of a group , which is defined by a series of subgroups

where is abnormal in , .

References

[1] B. Huppert, "Endliche Gruppen" , 1 , Springer (1967)


Comments

Nowadays, is mostly defined as . Instead of "solvable" also "soluble groupsoluble" is frequently used.

How to Cite This Entry:
Abnormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abnormal_subgroup&oldid=33149
This article was adapted from an original article by A.I. Kostrikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article