The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.

Let $G$ be a finite group. We show that the order of the subgroup generated by coprime ${\gamma }_{k} $-commutators (respectively, ${\delta }_{k} $-commutators) is bounded in terms of the size of the set of coprime ${\gamma }_{k} $-commutators (respectively, ${\delta }_{k} $-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words ${\gamma }_{k} $ and ${\delta }_{k} $ are concise.

The concepts of d- and nd-Frattini chief factors of a finite group are introduced. Their ingrainment into that of the extended Frattini dual subgroup becomes the natural dual to Frattini and supplemented chief factors. Not only does a dual of the strengthened form of the Jordan–Hölder theorem arise, but also the $p$-nilpotent radical becomes the intersection of the centralizers of the nd-Frattini chief factors. As a result, a class $\mathfrak{F}$ of groups is a full integrated local formation $\mathrm{LF}(f)$ if and only if each nd-Frattini chief factor in $G\in\mathfrak{F}$ is $f$-central.

Finite groups in which the Frattini subgroup of each proper normal subgroup is trivial, while the group itself has a nontrivial Frattini subgroup, are investigated. A direct result of this study leads to a classification of finite groups in which the Frattini subgroup of each proper subgroup is trivial, while the group itself has a nontrivial Frattini subgroup.