Group having a Sylow tower

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This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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Definition

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 p dividing the order of G, there is a unique quotient that is a p-subgroup and this group is isomorphic to a p-Sylow subgroup of G.

In other words, there exists a normal series:

1 = P_0 \le P_1 \le \dots \le P_r = G

such that for every p dividing the order of G, there exists a unique k such that P_k/P_{k-1} is isomorphic to a p-Sylow subgroup of G.

Relation with other properties

Stronger properties

Weaker properties

Incomparable properties

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 258, Chapter 7 (Fusion, transfer and p-factor groups), Section 7.6 (Elementary applications), More info