Abnormal subgroup

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.