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Stochastic Group

The Group of all nonsingular $n\times n$ Stochastic Matrices over a Field $F$. It is denoted $S(n,F)$. If $p$ is Prime and $F$ is the Galois Field of Order $q=p^m$, $S(n,q)$ is written instead of $S(n,F)$. Particular examples include

$\displaystyle S(2,2)$ $\textstyle =$ $\displaystyle \Bbb{Z}_2$  
$\displaystyle S(2,3)$ $\textstyle =$ $\displaystyle S_3$  
$\displaystyle S(2,4)$ $\textstyle =$ $\displaystyle A_4$  
$\displaystyle S(3,2)$ $\textstyle =$ $\displaystyle S_4$  
$\displaystyle S(2,5)$ $\textstyle =$ $\displaystyle \Bbb{Z}_4\times_\theta\Bbb{Z}_5,$  

where $\Bbb{Z}_2$ is an Abelian Group, $S_n$ are Symmetric Groups on $n$ elements, and $\times_\theta$ denotes the semidirect product with $\theta:\Bbb{Z}_4\to \mathop{\rm Aut}(\Bbb{Z}_5)$ (Poole 1995).

See also Stochastic Matrix


Poole, D. G. ``The Stochastic Group.'' Amer. Math. Monthly 102, 798-801, 1995.

© 1996-9 Eric W. Weisstein