arXiv subjects
Publications and source records attributed to Mohammad Zarrin.
In 1979, Herzog put forward the following conjecture: if two simple groups have the same number of involutions, then they are of the same order. We give a counterexample to this conjecture.
For any group G, let C(G) denote the intersection of the normal- izers of centralizers of all elements of G. Set C0 = 1. Define Ci+1(G)=Ci(G) = C(G=Ci(G)) for i ? 0. By C1(G) denote the terminal term of the ascending series. In this paper, we show that a finitely generated group G is nilpotent if and only if G = Cn(G) for some positive integer n.
A subset X of a group G is a set of pairwise non-commuting ele- ments if ab 6= ba for any two distinct elements a and b in X. If jXj ? jY j for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset of pairwise non-commuting elements and the cardinality of such a subset is denoted by !(G). In this paper, among other thing, we prove that, for each positive integer n, there are only finitely many groups G, up to isoclinic, with !(G) = n (with exactly n centralizers).
In this paper, we show that a locally graded group with a finite number m of non-(nilpotent of class at most n) subgroups is (soluble of class at most [log2(n)] + m + 3)-by-(finite of order $\le$ m!). Also we show that the derived length of a soluble group with a finite number m of non-(nilpotent of class at most n) subgroups, is at most [log2(n)] + m + 1.
Let G be a group and N be the class of nilpotent groups. A subset A of G is said to be nonnilpotent if for any two distinct elements a and b in A, ha, bi 62 N. If, for any other nonnilpotent subset B in G, |A| ? |B|, then A is said to be a maximal nonnilpotent subset and the cardinality of this subset (if it exists) is denoted by !(NG). In this paper, among other results, we obtain !(NSuz(q)) and !(NPGL(2,q)), where Suz(q) is the Suzuki simple group over the field with q elements and PGL(2, q) is the projective general linear group of degree 2 over the finite field of size q, respectively.
For any group G, let C(G) denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n. In this note, we prove that every finite Cn-group with n ? 21 is soluble and this estimate is sharp. Moreover, we prove that every finite Cn-group with |G| < 30n+15 19 is non-nilpotent soluble. This result gives a partial answer to a conjecture raised by A. Ashrafi in 2000.
We associate a graph $\mathcal{N}_{G}$ with a group $G$ (called the non-nilpotent graph of $G$) as follows: take $G$ as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of $\mathcal{N}_{G}$ and its induced subgraph on $G\backslash nil(G)$, where $nil(G)=\{x\in G | < x,y> \text{is nilpotent for all} y\in G\}$. For any finite group $G$, we prove that $\mathcal{N}_G$ has either $|Z^*(G)|$ or $|Z^*(G)|+1$ connected components, where $Z^*(G)$ is the hypercenter of $G$. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact we prove that a finite group $G$ is nilpotent if and only if the set of vertex degrees of $\mathcal{N}_G$ has at most two elements.