In this note alternate proofs of some basic results of nite group theory are presented. These notes contain a few new results. Our aim here is to give alternate and, as a rule, more short proofs of some basic results of nite group theory: theorems of Sylow, Hall, Carter, Kulako, Wielandt-Kegel and so on. Only nite groups are considered. We use the standard notation. (n) is the set of prime divisors of a natural number n and (G) = (jGj), where jGj is the order of a group G; p is a prime. A group G is said to be p-nilpotent if it has a normal p-complement. Given H < G, let HG = T x2G H x and H G be the core and normal closure of H in G, respectively. If M is a subset of G, then NG(M) and CG(M) is the normalizer and centralizer of M in G. Let sk(G) (ck(G)) denote the number of subgroups (cyclic subgroups) of order p k in G. Next, Epn is the elementary abelian group of order p n ; Cm is the cyclic group of order m; D2n, Q2n and SD2n are dihedral, generalized quaternion and semidihedral group of order 2 n , respectively. If G is a p-group, then n(G) = hx 2 G j o(x) p n i, f1(G) = hx p n j x 2 Gi, where o(x) is the order of x 2 G. Next, G 0 , Z(G), ( G) is the derived subgroup, the center and the Frattini subgroup of G; Sylp(G) and Hall (G) are the sets of p-Sylow and -Hall subgroups of G. We denote O (G) the maximal normal -subgroup of G. Let Irr(G) be the set of complex irreducible characters of G. If H < G