Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG (H) Ç Hg £ H for all g  Î G.A subgroup H of G is called a weakly H*-subgroup in G if there exists a subgroup K of G such that G=HK and H Ç K is an H-subgroup in G. We investigate the structure of the finite group G under the assumption that every cyclic subgroup of G of prime order p or of order 4 (if p=2) is a weakly H*-subgroup in G. Our results improve and extend a series of recent results in the literature.

A , nite group G, in which two randomly chosen subgroups H and K commute, has been classi, ed by Iwasawa in 1941. It is possible to de, ne a probabilistic notion, which \measures the distance" of G from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree gsd(G) for two arbitrary sublattices S(G) and T(G) of the lattice of subgroups L(G) of G. Upper and lower bounds for gsd(G) are shown and we study the behaviour of gsd(G) with respect to subgroups and quotients, showing new numerical restrictions.

Let $\gamma_{n}=[x_{1},\ldots,x_{n}]$ be the $n$th lower central word and $X_{n}(G)$ the set of $\gamma_{n}$-values in a group $G$. Suppose that $G$ is a profinite group where, for each $g\in G$, there exists a positive integer $n=n(g)$ such that the set $g^{X_{n}(G)}=\{g^{y}\,|\,y\in X_{n}(G)\}$ contains less than $2^{\aleph_{0}}$ elements. We prove that $G$ is a finite-by-nilpotent group.

In this paper, we give a complete proof of Theorem 4.1 (ii) and a new elementary proof of Theorem 4.1 (i) in [Li and Shen, On the intersection of the normalizers of the derived subgroups of all subgroups of a nite group, J. Algebra, 323 (2010) 1349-1357]. In addition, we also give a generalization of Baer's Theorem.

Let $c(G)$ denotes the number of cyclic subgroups of a finite group $G$. A group $G$ is said to be {\em $n$-cyclic}, if $c(G)=n$. In this paper, we classify all $11$-cyclic groups.

In the classical group theory there is an open question: Is every torsion free n-Engel group (for n  4), nilpotent? . To answer the question, Traustason [11] showed that with some additional conditions all 4-Engel groups are locally nilpotent. Here, we gave some partial answer to this question on Engel fuzzy subgroups...

Let n = [x1, : : :, xn] be the nth lower central word. Suppose that G is a pro, nite group where the conjugacy classes x n(G) contains less than 2@ 0 elements for any x 2 G. We prove that then n+1(G) has , nite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is , nite. Moreover, it implies that a pro, nite group G is , nite-by-nilpotent if and only if there is a positive integer n such that x n(G) contains less than 2@ 0 elements, for any x 2 G.

In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore, the Fundamental Homomorphism Theorem is established. Finally, it is proved that the class of all fuzzy convex lattice-ordered subgroups of a lattice-ordered group G forms a complete Heyting sublattice of the lattice of fuzzy subgroups of G.