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Paul Hedrick
2008-12-07
Math 152
Review of “The Stochastic Group,” by David G. Poole
“The Stochastic Group,” an article by David Poole found in The American
Mathematical Monthly, is an article that presents the idea of a new example of a group
that could be studied in a basic abstract algebra course. The article is rather short because
it is a simple presentation and description of the group along with a few lemmas and
accompanying proofs that are necessitated by Mr. Poole’s claims. To understand the
group he describes, the stochastic group, a couple definitions are set in the beginning of
the article. The most important definition for understanding the article is that of the
stochastic matrix over a field F, a square matrix with entries in F with the property that
the entries in each of its columns add up to 1. It will turn out to be similar to the concept
of general linear group and special linear subgroup matrices when it comes to counting
and determining their order, and some of the proofs for these concepts are nearly
identical to proofs we have covered in Math 152.
Before getting into too much detail about the material, though, I would like to
explain a little of the relevance of the article to what we are currently studying as well as
compare my view of the material with the author’s. Mr. Poole seems to purport that the
concept of the stochastic group is important and basic enough to be an essential part of
any undergraduate abstract algebra (or discrete mathematics…) course. It is a concept
that we have not studied in our curriculum, though, yet most of the important proofs and
definitions in the article are already so similar to concepts we have already studied that it
seems almost irrelevant to introduce the topic of stochastic groups other than in a purely
definitional context. I chose this article not because of its irrelevance but because the
author takes a strong standpoint on a concept that seems rather important, but I feel that
the article should be taken at face value. For a student in Math 152, the article could be
viewed as a supplement to the materials studied in Outline 8, for it goes into detail about
counting the order of stochastic groups and comparing it with general linear groups
(which we have already studied), but it also describes a couple isomorphisms of the
stochastic groups, which is similar work to what we did in Outline 8. One reason I almost
decided not to choose the article is that the material is rather dull—it does not involve
counting isomorphisms in popular dice or card games like many of the other suggested
articles do, but it certainly describes relevant material.
After defining what a stochastic matrix is, the article states and proves the
theorem that the set of nonsingular n  n matrices over a field F forms a group under
matrix multiplication. Once this is proven, the actual definition of a stochastic group can
be stated as the group of all nonsingular stochastic n  n matrices over a field F. It is
written as S(n,F) or S(n,q) if p is prime and F is the Galois field of order q = pm. Once
this definition is set, one can begin to discuss the actual structure of the stochastic group
and compare it to groups that we have already seen in Math 152. One example given in
the article is that the group S(2,2), the 2×2 stochastic matrices with entries in F2, is
isomorphic to Z2. This one is rather obvious since there are only two such matrices, the
identity [(1,0),(0,1)] and [(0,1),(1,0)] and only two elements in Z2. Another one that the
article does not state in detail but is rather easily shown is the isomorphism between
S(2,3) and S3, the symmetric group on 3 elements. There are 321 = 6 elements in S3,
and the isomorphic matrices of S(2,3) in row vector format are [(1,0),(0,1)], [(0,1),(1,0)],
[(2,1),(2,0)], [(2,0),(2,1)], [(0,2),(1,2)], and [(1,2),(0,2)]. These two are very simple
examples, but the article does not go into too much detail describing different symmetry
group and their respective isomorphic stochastic groups.
The next major theorem that is stated in the article is the theorem for counting the
order of a stochastic group S(n,q), and it turns out to be very similar to the method we
learned for counting the order of general linear groups. It is proposed that
|S(n,q)| = qn – 1 (qn – 1 – 1) (qn – 1 – q) … (qn – 1 – qn – 2).
This formula makes sense if the matrix is viewed column by column, counting the
number of possibilities for each one and multiplying them by each other. If U(Fn) is
defined as the set of all vectors in Fn whose components add up to 1 and the columns of
our concerned matrix A form an ordered basis {a1, a2, … , an}  Fn, then there are qn – 1
choices for a1 and (qn – 1 – 1) possibilities for a2 because any vector in the span of a1 must
be ruled out. This logic proceeds from a lemma that is proven in the article and is very
similar to what we learned in class. The only real difference is that there are qn – 1
possibilities for the first column instead of qn, and this is because they have to add up to
1, not simply have determinant = 1.
The article concludes with a categorization of stochastic groups using a concept
we have not studied in Math 152 called affine groups (not to be confused with affine
geometry). The article assumes some knowledge of this subject that a student of our level
will probably not have, so I will not describe the proof in detail. An affine group
Aff(n – 1, F) consists of all mappings : Fn – 1  Fn – 1 of the form (xT) = AxT + bT, x 
Fn – 1 where A is GL(n – 1, F) and b  Fn – 1, together with the operation of composition.
With this knowledge, the stochastic group can be categorized by one equation, such that
for any field F, S(n,F) is isomorphic to Aff(n – 1, F) for all n  2. Therefore, even a
student who does not completely understand the mathematics behind affine groups can
appreciate the fact that the concept of a stochastic group can be generalized. It is the
opinion of the author, then, that this group is important enough to be studied in any basic
undergraduate abstract algebra course. Whether or not that applies to Math 152 depends
on how much detail we will cover about group theory, but it is certainly an interesting
concept nonetheless.