“Ali Reza Zokayi”
IPM Positions 

Non Resident Researcher (nonresident), School of Mathematics
(2002  2003 ) 

Non IPM Affiliations 

Assistant Professor of Khajeh Nasir Toosi Technical University  
Research Interests 

Finite Groups  
Research Activities 

Characterization of finite groups through their element orders is one of the
most important problem in group theory after the full classification of finite
simple groups, which was completed in 1981. This problem was first introduced by
Prof. W.J. Shi. Soon later W.J. Shi investigated the finite groups in [SY] whose
element orders are of prime order except the identity element and he got the
interesting result: The alternating group A_{5} can be characterized
only by its element orders. At this time many researchers such as W.J. Shi, H.
Deng, V.D. Mazurov, A. Zavarnitsine, M.R. Darafsheh, A.R. Moghaddamfar and
myself, M.S. Lucido, R. Brandl, C.Y. Tang, S. Lipschutz, J. An etc, are working
on this problem.
Let G be a finite group. Denote by p_{e}(G) the set of all orders of elements in G. Obviously p_{e}(G) is a subset of the set N of natural numbers, and it is closed and partially ordered under divisibility. We sometimes let m(G) to be the set of maximal elements of (p_{e}(G), ). Let ? ? W ? N. Now we ask the following questions: (1) Is there a finite group G with p_{e}(G)=W ? (2) If the answer is affirmative then how many nonisomorphic groups exist with the above set of element orders? Certainly, W must have the following properties: (a) 1 ? W and (b) W must be closed and partially ordered with respect to divisibility. The conditions (a) and (b) are necessary but not sufficient, for example there does not exist any group G with p_{e}(G)=W = {1, 2, 3, 4, 5, 6, 7, 8, 9 }. In fact R. Brandl and W.J. Shi in [BS] have classified all finite groups whose element orders are consecutive integers and in this paper they have shown that if p_{e}(G)={1, 2, 3, ..., n }, for some group G, then n ? 8. Therefore, we consider a concrete group, say M, and set p_{e}(M)=W. Now it is evident that the answer of question (1) for such W is positive. So we try to answer the question (2). Let h(W) be the number of nonisomorphic classes of finite groups G such that p_{e}(G)=W. Given a group G, it is obvious that h(p_{e}(G)) ? 1. Now, we make the following definitions: A group G is called nondistinguishable if h(p_{e}(G))=?. A group G is called kdistinguishable if h(p_{e}(G))=k
< ?. Professor W.J. Shi has recently conjectured that h(p_{e}(G)) ? {1, 2, ?}, for any group G. It is well known that:
When n=1, we have Aut(PSL(2,p))=PGL(2,p) , and hence
In this research, we consider almost simple groups PGL_{2}(p), with 5
? p < 100, and we show that h(p_{e}(PGL_{2}(p)))
? {1, ?}. In fact we will
prove the following theorem: Proof of such Theorems, strengthen the Shi's conjecture.


Present Research Project at IPM 

Characterization of finite groups by their element orders  
Related Papers 
1.  M. R. Darafsheh, A. R. Moghaddamfar and A. R. Zokayi The characterization of PGL(2, p) for some p by their element orders Int. Math. Forum 1 (2006), 18251831 [abstract] 
2.  A. R. Zokayi (joint with A. R. Moghaddamfar and M. R. Darafsheh) A characterization of finite simple groups by the degrees of vertices of their prime graphs Algebra Colloq. 12 (2005), 431442 [abstract] 
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