Let G be a finite group and x be an element of G. Define SolG(x) as the set of all y is an element of G such that < x, y > is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely |N-G(< x >)| | | Sol(G)(x)| , where N-G(< x >) is the normalizer of < x >. Furthermore, we demonstrate that the conjecture holds in the special case where N-G(< x >) is a Frobenius group with kernel C-G(x), the centralizer of x and N-G(< x >) : C-G(x)| is of prime order. Finally, we will classify all finite simple groups G that contain an element x for which Sol(G)(x) is a maximal subgroup of order pq, where p and q are prime numbers.

Let 𝔑𝔦𝔩 be the class of nilpotent groups. This article explores the finiteness of meta and para-𝔑𝔦𝔩-Hamiltonian groups or their derived subgroups when these groups contain a soluble subgroup of finite index or a non-nilpotent (or insoluble) subgroup of finite order respectively.

Let 𝔑𝔦𝔩 be the class of nilpotent groups and G be a group. We call G a meta- 𝔑𝔦𝔩 -Hamiltonian group if any of its non- 𝔑𝔦𝔩 subgroups is normal. Also, we call G a para- 𝔑𝔦𝔩 -Hamiltonian group if G is a non- 𝔑𝔦𝔩 group and every non-normal subgroup of G is either a 𝔑𝔦𝔩 -group or a minimal non- 𝔑𝔦𝔩 group. In this paper we investigate the class of finitely generated meta- 𝔑𝔦𝔩 -Hamiltonian and para- 𝔑𝔦𝔩 -Hamiltonian groups.

Let G be a finite group, and let x be an element of G . Denote by Sol _G(x) the set of all y ∈ G such that the group generated by x and y is soluble. We investigate the influence of Sol _G(x) on the structure of G .

Let G be a finite p-group of odd order and let $$\nu _c(G)$$ be the number of conjugacy classes of non-normal cyclic subgroups of G. In this paper, among other results, we obtain a lower bound for $$\nu _c(G)$$ . More precisely, if $$|G'|=p^k$$ , for some integer k, then we prove that $$\nu _c(G)\ge k$$ .

Let G be a finite p-group. Assume that $$\nu (G)$$ and $$\nu _c(G)$$ denote the number of conjugacy classes of non-normal subgroups and non-normal cyclic subgroups of G, respectively. In this paper, we completely classify the finite p-groups with $$\nu _c=p$$ or $$p+1$$ for an odd prime number p. Also, we classify the groups G with $$\nu (G)=\nu _c(G)=p^i, i\ge 1$$ .

In this paper we show that a finite group G with Quaternion Sylow 2-subgroup is 2-nilpotent if, either 3 |G| or G is solvable and the order of its Sylow 2-subgroup is strictly greater than 16. Mathematical subject classification (2010). 20D99, 20E45. Manuscript received 7th October 2020, revised 11th October 2020 and 13th October 2020, accepted 13th October 2020.

A subgroup H of a group G is called a TI-subgroup if \(H^g\cap H=1\) or H for all \(g\in G\); and H is called quasi TI if \(\mathcal {C}_G(x)\le \mathcal {N}_G(H)\) for all non-trivial elements \(x\in H\). A group G is called (quasi CTI-group) CTI-group if every cyclic subgroup of G is a (quasi TI-subgroup) TI-subgroup. It is clear that TI subgroups are quasi TI. We first show that finite nilpotent quasi CTI-groups are CTI. In this paper, we classify all finite nilpotent CTI-groups.

‎‎For a finite group $G$‎, ‎let $nu_{nc}(G)$ denote the number of conjugacy classes of non-normal non-cyclic subgroups of $G$‎. ‎We characterize the finite non-nilpotent groups whose all non-normal non-cyclic subgroups are conjugate‎.

For a finite group G let v(G) denote the number of conjugacy classes of non-normal subgroups of G. The aim of this paper is to classify all the non-nilpotent groups with v(G) = 3.

For a finite group G let v(G) denote the number of conjugacy classes of non-normal subgroups of G. The aim of this paper is to classify all the non-nilpotent groups with v(G) = 3.