the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.

the rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. let $r$ be a ring. let $mathbb{a}(r)$ denote the set of all annihilating ideals of $r$ and let $mathbb{a}(r)^{*} = mathbb{a}(r)backslash {(0)}$. the annihilating-ideal graph of $r$, denoted by $mathbb{ag}(r)$  is an undirected simple graph whose vertex set is $mathbb{a}(r)...

in this paper, it was shown that , where  and , and , where  is not prime and , are od-characterizable.

The prime index graph π(G) of a finite group G is special type undirected simple whose vertex set subgroups G, in which two distinct vertices are adjacent if one has the other. Let p and q be primes. In this paper, we establish that cyclic p-group Zpn, abelian Zpn × Zq Zp always have graceful labeling without any condition on n using concept path or p-layer ladder size + 1.