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In a finite group  G , letπ  e   ⁢   (  G  ){\pi_{e}(G)}    be the set of orders of elements of  G , lets  k{s_{k}}    denote the number of elements of order  k  in  G , for eachk  ∈    π  e   ⁢   (  G  ){k\in\pi_{e}(G)}   , and then letnse  ⁡   (  G  ){\operatorname{nse}(G)}    be the unordered set{   s  k   :   k  ∈    π  e   ⁢   (  G  )}{\{s_{k}:k\in\pi_{e}(G)\}}   . In this paper, it is shown that if|  G  |   =   |    L  2   ⁢   (  q  )    |{\lvert G\rvert=\lvert L_{2}(q)\rvert}    andnse  ⁡   (  G  )    =   nse  ⁡   (    L  2   ⁢   (  q  )    ){\operatorname{nse}(G)=\operatorname{nse}(L_{2}(q))}    for some prime-power  q , then  G  is isomorphic toL  2   ⁢   (  q  ){L_{2}(q)}   .

Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order