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

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

2010-07-18

arxiv.org/abs/1007.2983v2

Mathematics

Group Theory

40 pages

Scientific paper

For an element g in a group X, we say that g has 2-part order 2^b if 2^b is the largest power of 2 dividing the order of g. Using results of Erd\H{o}s and Tur\'{a}n, and Beals et al., we give explicit lower bounds on the proportion of elements of the symmetric group S_n with certain 2-part orders. Some of these lower bounds are constant; for example we show that at least 23.5% of the elements in S_n, (n> 2) have a certain 2-part order. We derive related lower bounds on these proportions for finite classical groups in odd characteristic. In particular, we show that the proportion of odd order elements in the symplectic and orthogonal groups is at least C/n^{3/4}, where n is the Lie rank, and C is an explicit constant. Furthermore, we describe how these results can be used to analyze part of Yal\c{c}inkaya's Black Box recognition algorithm for finite classical groups in odd characteristic.

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