Group having a Sylow tower
This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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A group having a Sylow tower is a finite group that possesses a Sylow tower: a normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, and for each dividing the order of , there is a unique quotient that is a -subgroup and this group is isomorphic to a -Sylow subgroup of .
In other words, there exists a normal series:
such that for every dividing the order of , there exists a unique such that is isomorphic to a -Sylow subgroup of .
Relation with other properties
- Group having subgroups of all orders dividing the group order: For full proof, refer: Sylow tower not implies subgroups of all orders dividing the group order, Subgroups of all orders dividing the group order not implies Sylow tower