This paper proves: Let $\mathcal {F}$ be a saturated formation containing $\mathcal {U}$ . Suppose that $G$ is a group with a normal subgroup $H$ such that $G/H \in \mathcal {F}$ . (1) If all maximal subgroups of any Sylow subgroup of $F^*(H)$ are $c$ -supple- mented in $G$ , then $G \in \mathcal {F}$ ; (2) If all minimal subgroups and all cyclic subgroups with order 4 of $F^*(H)$ are $c$ -supplemented in $G$ , then $G \in \mathcal {F}$ .

The cyclic graph ΓG of a finite group G is as follows: take G as the vertices of ΓG and join two distinct vertices x and y if 〈x,y〉 is cyclic. In this paper, we investigate how the graph theoretical properties of ΓG affect the group theoretical properties of G. First, we consider some properties of ΓG and characterize certain finite groups whose cyclic graphs have some properties. Then, we present some properties of the cyclic graphs of the dihedral groups D2n and the generalized quaternion groups Q4n for some n. Finally, we present some parameters about the cyclic graphs of finite noncyclic groups of order up to 14.

The purpose of this paper is to study the influence of $c$-supplemented minimal subgroups on the $p$-nilpotency of finite groups. We obtain ‘iff' and ‘localized' versions of theorems of Itô and Buckley on nilpotence, $p$-nilpotence and supersolvability.

The purpose of this paper is to give some necessary and sufficient conditions for p-nilpotent groups. We extend some results, including the well-known theorems of Burnside and Frobenius as well as some very recent theorems. We also apply our results to determine the structure of some finite groups in terms of formation theory.