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A proper subgroup $H$ of a finite group $G$ is called a second maximal subgroup of $G$ if $H$ is a maximal subgroup of every maximal subgroup $M$ in $G$ with $H\le M$. In this paper, we investigate the structure of the finite group $G$ in which $C_G(x)$ is a second maximal subgroup for every non-central element $x$ in $G$, and prove that either $G$ is solvable or $G/Z(G)\cong \mathrm{\text{PSL}}(2,r^n)$, where $r=2$ with $n$ a prime or $r=3$ with $n$ an odd prime.

Let G be a finite group. A subgroup H of G is called weakly normal in G if H^g ≤ N_G(H) implies g ∈ N_G(H) for all g ∈ G. A finite group G is called an -group if all cyclic subgroups of G of order prime or 4 are weakly normal in G. In this paper, the structure of finite groups all of whose second maximal subgroups satisfy -property has been characterized.

A subgroup H of G is said to be self-conjugate-permutable if HH^x = H^xH implies H^x = H. A finite group G is called PSC-group if every cyclic subgroup of group G of prime order or order 4 is self-conjugate-permutable. In the paper, first we give the structure of finite group G, all of whose maximal subgroups are PSC-groups. Then we also classified that finite group G all of whose second maximal subgroups are PSC-groups.