Proportions of elements with given 2-part order in finite classical\n groups of odd characteristic

2010/07/18 by Simon D. Guest, Cheryl E. Praeger, Guest, Simon +1
Computer Science · Mathematics · #11E57 #20B30 #20P05 #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1007.2983

openalex publication_date 2010/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an element g in a group X, we say that g has 2-part order 2^a\nif 2^a is the largest power of 2 dividing the order of g. We prove lower\nbounds on the proportion of elements in finite classical groups in odd\ncharacteristic that have certain 2-part orders. In particular, we show that the\nproportion of odd order elements in the symplectic and orthogonal groups is at\nleast C/\ℓ^3/4, where \ℓ is the Lie rank, and C is an explicit\nconstant. We also prove positive constant lower bounds for the proportion of\nelements of certain 2-part orders independent of the Lie rank. Furthermore, we\ndescribe how these results can be used to analyze part of Yal ccinkaya's\nBlack Box recognition algorithm for finite classical groups in odd\ncharacteristic.\n

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