CHARACTER DEGREE GRAPHS OF ALMOST SIMPLE GROUPS A dissertation submitted to Kent ...

70 Pages·2014·0.35 MB·English

CHARACTER DEGREE GRAPHS OF ALMOST SIMPLE GROUPS
A dissertation submitted
to Kent State University in partial
fulfillment of the requirements for the
degree of Doctor of Philosophy
by
William M. Montanaro, Jr.
May, 2014
Dissertation written by
William Montanaro, Jr.
B.S., Kent State University, 2007
M.S., Kent State University, 2009
Ph.D., Kent State University, 2014
Approved by
Donald White, Chair, Doctoral Dissertation Committee
Stephen Gagola, Members, Doctoral Dissertation Committee
Mark Lewis
Deborah Smith
Jeffery Ciesla
Accepted by
Andrew Tonge, Chair, Department of Mathematical Sciences
Janis H. Crowther, Associate Dean, College of Arts and Sciences
ii
TABLE OF CONTENTS
ACKNOWLEDGMENTS .......................................................vi
Chapter
1. INTRODUCTION .........................................................1
2. BACKGROUND .......................................................... 6
3. NUMBER-THEORETIC LEMMAS ......................................10
4. OUTER AUTOMORPHISMS OF PSL (q) AND PSU (q2) ............... 15
3 3
5. CHARACTER DEGREE GRAPHS OF PSL (q) and PSU (q2) ...........36
3 3
5.1 PSL (q): One Outer Automorphism .............................. 37
3
5.2 PSL (q): Two Outer Automorphisms .............................40
3
5.3 PSL (q): Three Outer Automorphisms ............................43
3
5.4 PSL (q): Conclusions .............................................45
3
5.5 PSU (q2) .........................................................48
3
6. Sz(q) .....................................................................50
iii
7. OTHER GROUPS ....................................................... 60
BIBLIOGRAPHY ..............................................................62
iv
There are many people without whom this document would not have been made.
First among them are my advisor, Donald White. He has been an excellent teacher,
a humorous and intelligent conversationalist, and far more patient with me than I
deserve. Along with Dr. White, I would like to thank Mark Lewis and Stephen
Gagola for helping correct my numerous errors and teaching me the mathematics I
needed to write this.
I thank Deborah Smith and Jeffery Ciesla for serving on my defense committee,
DonaldDykesforintroducingmetoGroupTheory, FrankSandomierskiforreminding
me that I love mathematics, and Joyce Baier for convincing me that I did in fact want
to go to college in the first place.
IalsowanttothankBeverlyReed, AndrewTonge, VirginiaWright, MaryCordier,
and the mathematics graduate students at Kent State, including Cathy, Greg, Isaac,
Lena, Pat, Marina, and Monty. You have made my life and this process much easier.
I am thankful for my wife Molly, who has helped keep me from losing it and has
been patient even though this has taken me far too long to finish, and to my parents,
Kimberly, Richard, Bill, and Millie, for helping give me the opportunity to do this.
Finally, I want to thank the rest of my family. I have been incredibly lucky to have
you.
Thank you.
v
Chapter 1
Introduction
Since the late 1800s, representations have been used to study finite groups. A
complex representation X of a group G is a homomorphism from G to a group of
invertible matrices with entries from C. Representations encode a lot of information
about G. However, much of it is contained in the traces of these homomorphisms,
which are called characters.
Those characters that cannot be written as the sums of other characters are called
irreducible. The set of irreducible characters of G is Irr(G). It turns out that not
only is Irr(G) finite, but it also forms an orthonormal basis for the space of class
functions(functionsconstantonconjugacyclasses)ofG. Thevaluesoftheirreducible
characters on the identity element of G are integers called the character degrees of
G. The set of character degrees of G is denoted cd(G).
The number n of irreducible characters of G is equal to the number of conjugacy
classes of G. Let K and χ , 1 (cid:54) i,j (cid:54) n, be the conjugacy classes and irreducible
i j
characters respectively. The n×n matrix with entry a being the value χ takes on
ij j
K forms the character table of G. Given a character table, a lot of information about
i
G can be determined. For example, normal subgroups and nilpotency can be found.
Note that the character table does not in fact determine G uniquely (D and Q are
8 8
nonisomorphic but have the same character table).
In recent years, mathematicians have considered what information is contained
just in the character degrees of G. For example, Mark Lewis and Donald White have
classified the nonsolvable groups for which no character degree is divisible by the
square of an odd prime in [5].
1
2
One tool for studying the structure of character degrees is the character degree
graph, ∆(G). This graph has vertices that are the primes dividing an irreducible
character degree of G and two vertices p and q are adjacent if pq | χ(1) for some
χ ∈ Irr(G). An example of this being used is [4], in which Lewis and White classified
the nonsolvable groups whose degree graph is disconnected.
A simple group is one that has no nontrivial proper normal subgroups. To each
finite group, there corresponds a set of simple groups, called composition factors, that
are unique (up to isomorphism). Thus the finite simple groups are in some sense the
building blocks for all finite groups. The finite simple groups have been classified;
there are 18 (infinite) families and 26 sporadic simple groups. The classification of
finite simple groups is widely considered one of the crowning achievements of math-
ematics.
Using the classification, we can find information about the simple groups, then
extend those results to finite groups in general using information about their compo-
sition factors. The better the results are on the simple groups, the better the results
are for the general case.
The character degree graphs of the finite simple groups are known; see [10]. How-
ever, in many cases, more information is needed. In particular, for more recent work,
information on almost simple groups is useful. We say a group G is almost simple pro-
vided S (cid:54) G (cid:54) Aut(S), where S is a simple group and Aut(S) is the automorphism
group of S. The socle of a group G, Soc(G), is the (unique) subgroup of G generated
by the minimal normal subgroups of G. For an almost simple group G, the simple
group S is the socle of G. By Clifford theory, if N (cid:69)G, then character degrees of G
have the form σ(1) = r ·χ(1) where r | |G : N| and χ ∈ Irr(N). In particular, this
means ∆(S) is a subgraph of ∆(G). The interesting cases for the simple groups tend
to be the groups of smaller order; these have the fewest character degrees, whereas
3
those groups with more character degrees tend to have complete degree graphs.
In [11], White found the degree graphs of all almost simple groups G with socle
PSL (q), a projective special linear group of rank 1, where q = pf is a prime power.
2
Here we find the degree graphs of many almost simple groups G where the socle of
G is one of PSL (q), PSU (q2), a projective unitary group of rank 2 where q2 = pf,
3 3
or Sz(q), a Suzuki group where q = 22m+1, m (cid:62) 1. These groups were studied in
particular because their degree graphs are often incomplete. In addition, the explicit
character tables are known, which are necessary for the methods used here. As we use
the character tables of PSL (q) and PSU (q2) found in [8] extensively, we will also use
3 3
the notation of that paper for the characters and conjugacy classes of these groups,
and similarly we will use the notation of [9] when we discuss the Suzuki groups.
Interestingly, there are simple groups whose degree graph is not complete, but the
graph of their automorphism group is. In many of these cases, the full automorphism
groupisnotevennecessary. WeuseCliffordtheorytofindthestabilizersofirreducible
characters of the simple group in almost simple groups. Knowing the stabilizer of a
character gives information about it, e.g., if it induces irreducibly, and, if so, with
what degree. This is discussed further in Chapter 2.
The 18 infinite families of simple groups are formed by the cyclic groups of prime
order, the alternating groups, and the groups of Lie type, which are the groups we
will consider here. For the simple groups of Lie type, there are three types of outer
automorphisms: thediagonalautomorphisms,thefieldautomorphisms,andthegraph
automorphisms. In certain cases, some of these are inner. We will discuss the orders
of the outer automorphisms of PSL (q), PSU (q2), and Sz(q) below, and discuss the
3 3
orders of the outer automorphisms of the rest of the simple groups in Chapters 2
and 7.
The automorphisms of PSL (q) and PSU (q2) are easier to describe for SL (q) and
3 3 3
4
SU (q2). Then, as the centers of SL (q) and SU (q2) are invariant under these auto-
3 3 3
morphisms, they induce automorphisms on PSL (q) and PSU (q2). We will denote
3 3
the automorphisms of these groups the same way, omitting the “bar” notation.
The outer automorphism group of PSL (q) is generated as a semidirect product
3
by a diagonal automorphism δ of order (3,q−1), a field automorphism ϕ of order f,
where q = pf, and a graph automorphism γ of order 2. The diagonal automorphism
is induced on SL (q) by conjugation by a diagonal matrix in GL (q) of order q −1.
3 3
The field automorphism is induced by raising the matrix entries to the pth power,
and the graph automorphism by taking the inverse transpose of an element of SL (q).
3
The outer automorphism group of PSU (q2) is generated as a semidirect product
3
by a diagonal automorphism δ of order (3,q +1) and a field automorphism of order
f, where q2 = pf. The diagonal automorphism is induced on SU (q2) by conjugation
3
by a diagonal matrix in U (q2) of order (3,q+1). The field automorphism is induced
3
by raising the elements of the underlying field to the pth power.
The only outer automorphism of Sz(q) is the field automorphism, induced by
raising the matrix entries to the pth power.
In Chapter 2 we discuss some of the results our work depends on, including Clif-
ford theory. In Chapter 3, we prove number-theoretic lemmas that will be necessary.
In Chapter 4, we determine the effects of the outer automorphisms on various conju-
gacy classes and characters of PSL (q) and PSU (q2), and use these results to draw
3 3
conclusions about the character degrees of almost simple groups with socle PSL (q)
3
or PSU (q2). In Chapter 5 we write down explicitly the degree graphs of these almost
3
simple groups. In Chapter 6 we repeat the same steps for Sz(q), and in Chapter 7 we
note some conclusions for the degree graphs of almost simple groups whose socle is a
different simple group.
Chapter 2
Background
In this section, we will discuss various results and definitions we will be using.
We will follow much of the notation and development from chapters 5 and 6 of [3].
Many of the tools we will be using involve the way irreducible characters, particularly
their degrees, behave with normal subgroups. We begin by defining an induced class
function (recalling that a class function of a group G is a function that is constant
on the conjugacy classes of G). Let H (cid:54) G and let f be a class function of H. Then
1 (cid:88)
fG(g) = f∗(xgx−1),
|H|
x∈G
where f∗(h) = f(h) for h ∈ H and f∗(y) = 0 for y ∈/ H. Note that fG is a class
function on G and fG(1) = |G : H|f(1). Also, if f is a character of H, then fG is a
character of G. It does not follow that if f is irreducible, then fG is also.
Now we restrict our attention to the situation where we have a normal subgroup,
N (cid:69)G. Let f be a class function of N and g ∈ G. Define fg by fg(x) = f(gxg−1);
we call fg conjugate to f in G. If f is a character of N, then fg is a character of N.
Furthermore, G permutes Irr(N) by g : χ (cid:55)→ χg. Here, N acts trivially; thus G/N
permutes Irr(N).
This brings us to Clifford’s Theorem (6.2 in [3]):
Theorem 2.1. Let N (cid:69)G and χ ∈ Irr(G). Let θ be an irreducible constituent of χ
N
and suppose θ , 1 (cid:54) i (cid:54) t, are the distinct conjugates of θ in G. Then
i
t
(cid:88)
χ = [χ ,θ] θ .
N N i
i=1
5