en تخصصي Special پژوهشي Research paper <div style="text-align: center;"></div> <div style="text-align: justify;">In 2016, the second and third authors introduced the notion of 2-Engelizer of the element x in a given group G and denoted the set of all 2-Engelizers in G by E²(G). They also constructed the possible values of |E²(G)| [Bull. Korean Math. Soc., 53(3), (2016), 657-665]. In the present paper, we classify all non 2-Engel finite groups G, when |E²(G)| = 4, 5.</div> 2-Engelizer subgroup, 2-Engel element, 2-Engel group. 95 105 http://ijmsi.ir/browse.php?a_code=A-10-2629-1&slc_lang=en&sid=1 Raheleh Khoshtarash rakho86@gmail.com 100319475328460010695 100319475328460010695 No Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran Mohammad Reza Rajabzadeh Moghaddam rezam@ferdowsi.um.ac.ir 100319475328460010696 100319475328460010696 Yes Department of Mathematics, Khayyam University, Mashhad, Iran Mohammad Amin Rostamyari m.a.rostamyari@khayyam.ac.ir 100319475328460010697 100319475328460010697 No Department of Mathematics, Khayyam University, Mashhad, Iran