On the Fitting height of a finite group

Boundedly generated subgroups of finite groups

“…Thus, we observe here a very interesting phenomenon that in a sense the structure of some 2-generated subgroup of a finite group should be as complex as that of the whole group. A further illustration for this is the theorem obtained in [13] that in particular implies that a finite group G is soluble and has Fitting height h if and only if every 2-generated subgroup of G is soluble and has Fitting height h. It is natural to ask what other properties of G can be detected by looking at subgroups with small number of generators. In the present paper we address the problem for such important characteristics of a group G as the set of all prime divisors of the order of G (denoted by .G/), the set of composition factors (up to isomorphism), the exponent, the prime graph, or the spectrum of the group G.…”

On the length of a finite group and of its 2-generator subgroups

“…We will require the following results obtained in [7] and [10], respectively. The proof of Theorem 1.3 is now straightforward.…”

“…A deep theorem of Thompson says that G is soluble if and only if every 2-generator subgroup of G is soluble [11] (see also Flavell [4]). A number of recent results reflecting the phenomenon that properties of a finite group are determined by its boundedly generated subgroups can be found in [10,9,2].…”

Commutators in residually finite groups

“…The proof also relies on the classification of finite simple groups as well as on the Lie-theoretical techniques that Zelmanov created in his solution of the Restricted Burnside Problem. We also use our recent result that a finite group G is soluble with Fitting height at most h if and only if every pair of conjugate elements generates a subgroup with that property [24].…”

“…Set h = h(n). By [24] G contains two conjugate elements a, b such that h( a, b ) ≥ h + 2. Put H = a, b .…”