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Подгруппа Виландта конечной группы $G$ определяется как $w(G)={\bigcap\limits_{H \mathbin{{\lhd} {\lhd}} G}N_{G}(H)}$. В статье эта подгруппа вычислена для некоторых конечных групп.

Let Γ(V, E) be a graph. The Cayley graph of group (G,∗) and a subset S is a graph denoted by C a y(G, S), it is with vertex set V(C a y(G, S)) = G and two vertices x, y ∈ G are join if and only if y = x s ∈ S for some s ∈ S. In this paper, we obtain characteristics of Cayley graphs under some graph operations, Graph direct product, Graph tenser product, Graph strong product when G isomorphic to Cyclic group C n and Dihedral Group D 2n .

Let τ(n) is the number of all divisors of n and σ(n) is the number of summation of all divisors n, Cavior, presented the number of all subgroups of the dihedral group is equal by τ(n) + σ(n). We in this paper determines a formula for the number of subgroups, normal and cyclic subgroups of the group G = D 2n × C p = 〈a, b, c|a n = b 2 = c p , b a b = a −1, [a, c] = [b, c] = 1〉, where p is an odd prime number.

Studying the orbit of an element in a discrete dynamical system is one of the most important areas in pure and applied mathematics. It is well known that each graph contains a finite (or infinite) number of elements. In this work, we introduce a new analytical phenomenon to the weighted graphs by studying the orbit of their elements. Studying the weighted graph's orbit allows us to have a better understanding to the behaviour of the systems (graphs) during determined time and environment. Moreover, the energy of the graph’s orbit is given.