Odd-order and normal rank two for all primes implies Sylow tower
This article states and (possibly) proves a fact that is true for certain kinds of odd-order groups. The analogous statement is not in general true for groups of even order.
View other such facts for finite groups
This article describes a result about finite groups where the isomorphism types of the Sylow subgroup (?)s implies certain properties of the whole group. Note that all -Sylow subgroups for a given prime are conjugate and hence are isomorphic, so the statement makes sense.
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Suppose is a finite group of odd order. Further, suppose that for every prime dividing the order of , and every -Sylow subgroup of , the normal rank of is at most two. Then, has a Sylow tower.
Breakdown for even order
The result fails if we allow groups of even order. The smallest counterexample is the symmetric group of degree four, where both Sylow subgroups have normal rank at most two, but the group does not possess a Sylow tower.