What is it about?
One of the important problems in finite groups theory is the characterization of groups by a specific property. Researchers used different ways for the characterization of groups. For example, characterization groups by elements order, the elements with the same order, the number of order components, the number of Sylow $p$-subgroups for each prime $p$ etc. In this paper, we proved that 2-dimensional projective linear groups characterizable by the set of the numbers of elements of the same order.
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Why is it important?
If a group $G$ characterizable by the set of the numbers of elements of the same order, then every group the same order type with $G$ is isomorphic to $G$.
Perspectives
Characterization of other simple groups by the set of the numbers of elements of the same order is an open problem. Researchers can work on other simple groups.
Alireza Khalili
Farhangian University, Tehran, Iran
Read the Original
This page is a summary of: Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order, journal of Groups complexity cryptology, October 2018, De Gruyter,
DOI: 10.1515/gcc-2018-0011.
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