REU 2001
I was priviliged to be part of an NSF grant to conduct a Summer Research Experience Undergraduates. This is a joint program between Clarkson and SUNY Potsdam. During the summer of 2001 I was advising a group of four students in the investigation of Root Groups. We hope you will enjoy the pictures and brief comments that illustrate our summer program.
Clarkson/Potsdam REU 2001 Group Photo
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Doug Van Nort (Group Leader) |
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Joshua Wood |
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Heather |
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Alex Rogalski |
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Dr. Blair Madore |
In the integers, the additive square root of 4 is 2, 2 has (additive) square root 1, and 1 has no square root. In the rational numbers this chain of square roots would continue indefinitely from 4 to 2 to 1 to ½ to ¼ to …
Question 1: Can you find a (non-trivial) group where one element has 2^n th roots for all natural numbers n but no infinite chain of square roots?
Define a class of groups, we call Root Groups, on the infinite Cartesian product Z x Za(1) x Za(2) x Za(3) … where all but finitely many co-ordinates are zero. Define an operation on this set by addition in the Z co-ordinate and in the Za(i) co-ordinate, use addition but with a carry of one into the Z co-ordinate. In the case a(i)=2i we obtain the group described by King in [1], which is the answer to our Question 1. I was able to use this group in the field of Ergodic Theory to construct a measure preserving transformation with all 2n th roots but no infinite square root chains [2], in answer to a question of King.
Of course one cannot help but notice the root groups are a very interesting class of groups. They are (at most) countable generated and Abelian. They are composed of groups familiar to any undergraduate, the integers and finite cyclic groups, yet pieced together with a simple and unusual operation. There are many interesting questions surrounding these groups.
Question 2: What is the structure of the finitely generated Root Groups?
If a(i) is a finite sequence the group is finitely generated Abelian and thus standard structure theory applies. Yet for any given sequence one would have to perform a Smith Normal Form calculation (see [3]). Is there no way to predict in advance the structure based upon the sequence a(i)? For instance, our team rediscovered quickly that the Root Group Z o Za(1) is isomorphic to Z and that Z o Za(1) o Za(2) is isomorphic to Z Å Zd where d=gcd(a(1),a(2)). This pattern should generalize.
Question 3: What is the structure of the infinitely generated Root Groups?
The standard structure theory also extends to infinitely generated groups (Fuchs[3]) though is much more demanding.
Question 4: Are the Root Groups a rich enough class of groups to include all finitely generated Abelian groups with an element of infinite order? All countably generated Abelian groups?
Question 5: What root sequences are possible for an element of infinite order in an Abelian group? In a non Abelian group? For instance an element with roots of order 2 and 3 automatically has a root of order 6 in an Abelian group.
In the field of Ergodic theory it is becoming increasingly clear that one cannot study measure preserving transformations without studying the group of transformations they commute with. The answers to these questions about Root Groups are interesting in and of themselves but could also lead to some very exciting examples in Ergodic theory.
References:
[1] J. King, "The generic transformation has roots of all orders", to appear in Colloquium Mathematicum.
[2] B. Madore, "Rank-one group actions with simple mixing Z
subactions" Ph. D. thesis,
[3] L. Fuchs, "Abelian Groups", Pergamon Press,
[4]
T. Hungerford, "Algebra",
[5] B. Madore, “Rank one group
actions with simple Z-subactions”.
Let ai be a sequence of natural numbers. Let Za(i) be
represented by {0,1,…a(i)-1} with addition mod a(i). Define a Root Group
G, on the subset of the infinite Cartesian product Z x Za(1) x Za(2)
x Za(3) … consisting of elements with only finitely many
non-zero co-ordinates.
Denote this set by G=Z o Za(1) o Za(2) o Za(3)
… and define an operation · on G
by
(b, b1, b2, …) · (c, c1, c2, … ) = (b+c+CARRY, b1+c1 mod a1, b2+c2 mod a2, … )
Where CARRY is the number of i such that bi+ci >=
a(i). Informally this is addition in the Z co-ordinate and in the Za(i)
co-ordinate use addition but with a carry of 1, if necessary, into the Z
co-ordinate. For example in Z o Z2 o Z5,
(-7,1,4) · (-1,1,3) = (-7-1+2, 1+1 mod
2, 4+ 3 mod 5) = (-6,0,2).
Question: What is the structure of the finitely generated Root Groups?
If a(i) is a finite sequence the group is finitely generated Abelian and thus the Fundamental Theorem of finitely Generated Abelian Groups applies. Yet for any given sequence one would have to perform a Smith Normal Form calculation (see [2]). Is there no way to predict in advance the structure based upon the sequence ai?
We considered simple cases of this question first, discovering two important
and basic facts:
1. The Root Group Z o Za(1) is isomorphic to Z
2. Z o Za(1) o Za(2) is isomorphic to Z Å Zd where d=gcd(a(1),a(2)).
We also generalized the definition of root group to include groups where the
carry could be a natural number m different from 1. For example, in the group Z
o3 Z5 we have (1,2) ·
(-2, 4) = (1-2+3, 2+4 mod 5)=(2,1). This leads to our third important fact.
3. Z oa(1) Za(2) is isomorphic to Z o Za(1) o
Za(2)
Using this three facts in a clever way that involved analyzing the specific isomorphism used we were able to find a general structure theorem for finitely generated root groups. Let (x,y) denote the greatest common divisor of x and y and let [x,y] denote their lowest common multiple.
Theorem: Z o Za(1) o Za(2) o Za(3) … o Za(n) is isomorphic to Z Å Z(a(1),a(2)) Å Z([a(1),a(2)],a(3)) … o Z([a(1),a(2),…a(n-1)],a(n))
Using this theorem it is easy to show that root groups are a very specific class of finitely generated Abelian groups.
Corollary: Every finitely generated Abelian group with containing only one copy of the integers is a root group.
With Dr. Miller’s help a paper on this topic was completed and published [3]. Later Dr. Miller with help from Dr. Madore published an article on how to use our groups in projects for an abstract algebra class[4].
References:
[1] L. Fuchs, "Abelian Groups",
Pergamon Press,
[2] T. Hungerford, "Algebra",
[3] B. Madore, H. McDonough, C. Miller, D. Van Nort, A. Rogalski, and J. Wood, “Structure theory for Carry Groups”, Pi Mu Epsilon Journal, Vol. 12 (Fall 2004) 1, 17-25.
[4] B.Madore and C. Miller, “Carry Groups: Abstract Algebra Projects”, PRIMUS, Vol. XIV (Fall 2004) 3, 258-268.
We hold a number of exciting events during our REU so that students and faculty can get to know each other better outside the school environment. Surprisingly some of the best math discussions happen in these informal atmospheres. What follows are photos from some of these activities.
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June 23, 2001 Bottom: Matt, Heather, Cat, Sarah and Barbara |
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June 19 and 25 Canoeing on With the help of visitor Ruben Martinez of MSU, our group tried canoeing before taking the larger REU group canoeing for an afternoon. This area of incredible beauty is right next to campus. SUNY Potsdam and Clarkson both have boathouses on the river with equipment available for loan/rent. Right: Cole with Blair and Ruben |
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June 30 Mount Marcy and What a fabulous day hiking in high peaks.. Dr. Mahdavi's A-team made it
within a few hundred feet of the summit before a thunderstorm hit. The B-team
made it up to Clockwise from right: Mt Marcy is in the clouds as viewed form Marcy Dam; back at Marcy Dam soaking our feet; Cat, Hester, Barbara and Ryan at Indian falls; the whole crew loading up to go home; Indian Falls. |
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July 16 Picnic at Lampson Falls Just inside the Clockwise from right: Base of the falls; Heather, Hester, Cole and Dr. Madore; View from the top |
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"I chose the Clarkson-Potsdam REU program because I liked the math that was going to be studied. I am now more interested in math and more determined to go to graduate school in math because of this experience. Additionally, I feel that I could now conquer any isomorphism!" -- Heather McDonough |
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"I wanted some nice math experience and a chance to avoid a mindless
job. I chose the |
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"I was helped in solidifying my knowledge of group theory, in particular finitely generated abelian groups. I expected my group to be dysfunctional as last years, but was pleasantly surprised to find it untrue." -- Doug Van Nort |
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"A REU in mathematics is an excellent opportunity for any aspiring graduate math student. It is a good test for doing graduate work as you work with a small number of people and bounce ideas/concepts off each other to see if you can solve a given problem. It also broadens your horizon as you work in an area of mathematics that you have no extensive knowledge of. I would highly recommend this REU to other students as it was a great eye opener to new areas of math." -- Joshua Wood |
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Clarkson/Potsdam REU 2001 group photo
Particpants:
Tall: James Worthington, Ryan Belk, Alex Rogalski, Doug Van Nort
Medium: Dr. Tino Tamon, Eli Drake, Dr. Biman Das, Dr. Lou Kauffman, Amarda, Mike and April Siegler, David Lindsay, Hester Graves, Cat Vanderwaart, Sarah Good, Joshua Wood, Dr. Blair Madore, Dr. Joel Fosiy
Short: Dr. Kazem Mahdavi, Matt Day, Heather Mcdonough, Barbara Chervenka
For more information about the Clarkson-Potsdam REU see the offical home page.
For questions about this page contact me at madorebf@potsdam.edu