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FINDING INVOLUTIONS WITH SMALL SUPPORT

Part of: Permutation groups Representation theory of groups Probabilistic methods in group theory

Published online by Cambridge University Press:  11 January 2016

ALICE C. NIEMEYER  and
TOMASZ POPIEL
ALICE C. NIEMEYER*
Affiliation:
Lehrstuhl B für Mathematik, RWTH Aachen University, Pontdriesch 10–16, 52062 Aachen, Germany email alice.niemeyer@math.rwth-aachen.de
TOMASZ POPIEL
Affiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia email tomasz.popiel@uwa.edu.au
*
✉alice.niemeyer@math.rwth-aachen.de
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Abstract

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We show that the proportion of permutations $g$ in $S_{\!n}$ or $A_{n}$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^{{\it\varepsilon}}\rceil$ is at least a constant multiple of ${\it\varepsilon}$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^{{\it\varepsilon}}\rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2\rfloor ^{{\it\varepsilon}}\rceil$ for a symplectic or orthogonal group.

Keywords

symmetric groupalternating groupclassical groupproportion of elementsinvolution

MSC classification

Primary: 20D06: Simple groups
Secondary: 20P05: Probabilistic methods in group theory 20B30: Symmetric groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 43 - 47
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

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