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In the past thirty years, many authors investigated some quantitative characterizations of finite groups, especially finite simple groups, such as quantitative characterizations by group order and element orders, by the set of lengths of conjugacy classes, by dimensions of irreducible characters, etc. In this article the projective special linear group is characterized by its order and one special conjugacy class size, where p is a prime. This work implies that Thompson’s conjecture holds for . MSC: 20D08, 20D60.

Recognizing L 2 ( p ) by its order and one special conjugacy class size