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Stochastic Group


The group of all nonsingular n×n stochastic matrices over a field F. It is denoted S(n,F). If p is prime and F is the finite field of order q=p^m, S(n,q) is written instead of S(n,F). Particular examples include

S(2,2)=Z_2
(1)
S(2,3)=S_3
(2)
S(2,4)=A_4
(3)
S(3,2)=S_4
(4)
S(2,5)=Z_4×_thetaZ_5,
(5)

where Z_2 is an Abelian group, S_n are symmetric groups on n elements, and ×_theta denotes the semidirect product with theta:Z_4->Aut(Z_5) (Poole 1995).


See also

Stochastic Matrix

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References

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

Referenced on Wolfram|Alpha

Stochastic Group

Cite this as:

Weisstein, Eric W. "Stochastic Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StochasticGroup.html

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