Rational Schur Rings over Abelian Groups

90 Pages·2016·0.58 MB·English

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Theses and Dissertations
2008-07-08
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Brent L. Kerby
Brigham Young University - Provo
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Kerby, Brent L., "Rational Schur Rings over Abelian Groups" (2008). Theses and Dissertations. 1491.
https://scholarsarchive.byu.edu/etd/1491
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Rational Schur Rings over Abelian Groups
by
Brent Kerby
A thesis submitted to the faculty of
Brigham Young University
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Mathematics
Brigham Young University
August 2008
Copyright (cid:13)c 2008 Brent Kerby
All Rights Reserved
BRIGHAM YOUNG UNIVERSITY
GRADUATE COMMITTEE APPROVAL
of a thesis submitted by
Brent Kerby
This thesis has been read by each member of the following graduate committee and
by majority vote has been found to be satisfactory.
Date Stephen Humphries, Chair
Date Darrin Doud
Date William E. Lang
BRIGHAM YOUNG UNIVERSITY
As chair of the candidate’s graduate committee, I have read the thesis of Brent Kerby
initsfinalformandhavefoundthat(1)itsformat, citations, andbibliographicalstyle
areconsistentandacceptableandfulfilluniversityanddepartmentstylerequirements;
(2) its illustrative materials including figures, tables, and charts are in place; and
(3) the final manuscript is satisfactory to the graduate committee and is ready for
submission to the university library.
Date Stephen Humphries
Chair, Graduate Committee
Accepted for the Department
William E. Lang
Graduate Coordinator
Accepted for the College
Thomas Sederberg, Associate Dean
College of Physical and Mathematical Sciences
ABSTRACT
Rational Schur Rings over Abelian Groups
Brent Kerby
Department of Mathematics
Master of Science
In 1993, Muzychuk showed that the rational S-rings over a cyclic group Z are in
n
one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently,
with sublattices of the lattice of subgroups of Z . This idea is easily extended to show
n
that for any finite group G, sublattices of the lattice of characteristic subgroups of G
give rise to rational S-rings over G in a natural way. Our main result is that any finite
group may be represented as the automorphism group of such a rational S-ring over
an abelian p-group. In order to show this, we first give a complete description of the
automorphism classes and characteristic subgroups of finite abelian groups. We show
that for a large class of abelian groups, including all those of odd order, the lattice of
characteristic subgroups is distributive. We also prove a converse to the well-known
result of Muzychuk that two S-rings over a cyclic group are isomorphic if and only if
they coincide; namely, we show that over a group which is not cyclic, there always
exist distinct isomorphic S-rings. Finally, we show that the automorphism group of
any S-ring over a cyclic group is abelian.
ACKNOWLEDGMENTS
I would like to thank Darrin Doud for the use of his computer to carry out
MAGMA computations, without which many results in this thesis would not have
been discovered.
Contents
1 Basic definitions and elementary results 1
2 Central and rational S-rings 11
3 Isomorphisms and Automorphisms of S-rings 14
4 Automorphism Classes of Abelian Groups 20
5 Characteristic Subgroups of Abelian Groups 28
6 Main Theorem 49
7 Proof of Birkhoff’s Theorem 56
8 Automorphisms of S-rings over cyclic groups 59
A Appendix: MAGMA code 71
References 81
vii
List of Tables
1 Number of S-rings over Z , n < 192, for coefficient ring R of characteristic 0 4
n
2 Number of S-rings over non-cyclic groups of order ≤ 20 for coefficient ring R of characteristic
3 Characteristic subgroups of G = Z ×Z for odd prime p . . . . . . 31
p p3
4 Characteristic subgroups of G = Z ×Z ×Z for odd prime p . . . 32
p p3 p5
5 Characteristic subgroups of Z ×Z . . . . . . . . . . . . . . . . . . . 38
2 8
6 Number of characteristic subgroups of Z ×Z ×Z ×···×Z . . 41
2 22 23 2n
7 Characteristic subgroups of Char(Z ×Z ) and Char(Z ×Z ×Z ), p 6= 2 45
p2 p5 p p2 p4
8 Characteristic subgroups of Char(Z ×Z ) and Char(Z ×Z ×Z ), p = 2 45
p2 p5 p p2 p4
9 Characteristic subgroups of G = Z ×Z . . . . . . . . . . . . . . . . 47
4 64
viii
1 Basic definitions and elementary results
In this section, fix a commutative ring R with unity. Except where stated otherwise,
all groups considered will be finite.
If G is a group, the group algebra of G with coefficients in R is denoted RG. If
C is a subset of G, then we define C ∈ RG by C = g, and call C a simple
g∈C
quantity of the group algebra RG. Given a subset CP⊆ G and an integer m, we
define C(m) = {gm : g ∈ C}. For any x ∈ RG, where x = r g, we define
g∈G g
x(m) = r gm. Given a subset X ⊆ RG, we denote the R-sPubmodule generated
g∈G g
by X asPRX, i.e.,
k
RX = { r x : k ∈ N,r ∈ R,x ∈ X}.
i i i i
i=1
X
Definition 1.1. Let G be a finite group. An R-submodule S of the group algebra
RG is called a Schur ring (or S-ring) over G if there are disjoint nonempty subsets
T ,...,T ofGsuchthatS = R{T ,...,T }(i.e.,T ,...,T spanS asanR-module),
1 n 1 n 1 n
with the following properties,
(i) T T ∈ S for all i,j ∈ {1,...,n}.
i j
(−1)
(ii) For every i there is some j such that T = T .
i j
(iii) T = {1}, and G = T ∪T ∪···∪T .
1 1 2 n
The sets T ,...,T are called basic sets of S and are said to form a Schur partition
1 n
of G. The corresponding T ,...,T are called basic quantities of S. If S satisfies
1 n
condition (i) and (ii) but perhaps not (iii) then S is called a pseudo S-ring (or PS-
ring).
Note that condition (i) ensures that S is closed under multiplication, so that S is
in fact a subalgebra of RG. It is easy to see that if S is a PS-ring, then the collection
1