The influence of $$IC\Phi$$-subgroups on the structure of finite groups

On finite groups with given $ICΦ$-subgroups

“…The goal of the present paper is to further study how the structure of a group is influenced by its ICΦ-subgroups. Our results show, together with [5] and [12], that one often gets very rich information about a group when some of its subgroups are assumed to be ICΦ-subgroups.…”

“…Theorem 1.3 is covered by Theorem 1.4. Also, the proof of Theorem 1.4 given here is shorter than the proof of Theorem 1.3 given in [5].…”

“…This concept was introduced by Gao and Li in [5] and further investigated by the author in [12]. The papers [5] and [12] contain results that constrain the structure of a group G under the condition that some given subgroups of G are ICΦ-subgroups of G. The following theorem is the main result of [12]. The goal of the present paper is to further study how the structure of a group is influenced by its ICΦ-subgroups.…”

“…Given a group G and a subgroup H of G, we say that H is an ICΦ-subgroup of G provided that H ∩ [H, G] ≤ Φ(H). This concept was introduced by Gao and Li in [5] and further investigated by the author in [12]. The papers [5] and [12] contain results that constrain the structure of a group G under the condition that some given subgroups of G are ICΦ-subgroups of G. The following theorem is the main result of [12].…”