**Nadimpalli - Advanced Abstract Algebra

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Advanced Abstract Algebra
Shivam Nadimpalli
Last updated: June 1, 2018

Advanced Abstract Algebra Spring 2018
Contents
0 Hello! 3
I Topics in Group Theory 4
1 Groups Actions 5
1.1 Definition of a group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Orbitsandstabilizers................................ 6
1.3 Orbit-Stabilizer Theorem and Burnside’s Lemma . . . . . . . . . . . . . . . . 8
1.4 SylowTheorems .................................. 9
2 The Transfer Homomorphism & Applications 11
2.1 Thetransferfunction................................ 11
2.2 Properties of the transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Making the transfer prettier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 The transfer homomorphism in action . . . . . . . . . . . . . . . . . . . . . . 13
3 Solvable and Nilpotent Groups 19
3.1 Revisiting linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Solvable and Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Twospecialsubgroups............................... 23
3.4 An application to finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Free Groups 28
4.1 Construction of free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Cayleygraphs.................................... 29
4.3 Funwithfreegroups................................ 30
II Representation Theory of Finite Groups 34
5 Basic Notions 35
5.1 What is a representation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Maps between representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Maschke’s Theorem and Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . 37
6 Characters and Burnside’s Theorems 41
6.1 Whatisacharacter?................................ 41
6.2 Orthogonality relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 AlgebraicIntegers ................................. 45
6.4 Two Theorems of Burnside . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7 More Character Theory 50
7.1 Inducedcharacters ................................. 50
7.2 Frobeniusgroups.................................. 53
7.3 The Second Orthogonality Relation . . . . . . . . . . . . . . . . . . . . . . . . 55
7.4 MoreApplications ................................. 56
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Advanced Abstract Algebra Spring 2018
0 Hello!
These notes are based on the Advanced Abstract Algebra course taught by Dr. P´eter
Hermann, as a part of the Budapest Semesters in Mathematics, Spring 2018. The course
attempts to give an idea of some basic methods in finite and infinite group theory. All
errors in these notes are my responsibility; please email any comments or typos to nadim-
[e-mail].
Notation
Below we keep a running track of notation used in class. Occasionally, especially in the
section on representation theory, we might write gφinstead of φ(g).
O(g) The order of a group element g
G(x) Orbit of xunder action of G
GxStabilizer of xunder action of G
CG(g) Centralizer of gin G
Z(G) Center of G
CaConjugacy class of ain G
NG(H) Normalizer of H≤Gin G
Sylp(G) Sylow p-subgroups of G
τG→ATransfer homomorphism from Gto abelian subgroup Aof finite index
[a, b] Commutator of a, b =a−1b−1ab
[A, B]h[a, b]|a∈A, b ∈Bi
G0Commutator subgroup of G; equivalently [G, G]
φ(G) Frattini subgroup of G
F(G) Fitting subgroup of G
F(X) Free group generated by X
Γ(G;X) Directed, colored Cayley graph of group Ggenerated by X
Hom(V, W ){f:V→W|fis a linear map}
End(V) Equivalent to Hom(V, V )
GL(V) The group of invertible endomorphisms of V
GLn(C) Invertible n×nmatrices with entries from C
χWill usually stand for the character of a representation
χGInduced characters, see corresponding section for more
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Part I
Topics in Group Theory

Advanced Abstract Algebra Spring 2018
1 Groups Actions
Group actions are probably the most central topic in group theory: if a group Gacts on a set
X, then it captures some information about the symmetries of X. Almost every important
topic that we will talk about in this course will be formulated in the language of group
actions.
As a motivating example, suppose we want to count the number of its symmetries, namely
the number of edge preserving bijections on its vertex set.
Instead of attempting to count the number of valid transformations, it is much easier to
study a group that “acts” on its vertices.
We now give the precise definition of a group action, and introduce related concepts.
1.1 Definition of a group action
Definition 1.1.1 (group action)
Let X6=∅be a set, and Ga group. Then Gacts on Xif:
1. (∀α∈X)(∀c∈G) : c(α)∈X
2. (∀α∈X)(∀c, d ∈G) : cd(α) = c(d(α))
3. (∀α∈X) : e(α) = α
Example 1.1.2
Consider the trivial action of a group Gon X: (∀α∈G)(∀x∈X) : α(x) = x.
We now look at an alternative definition of group actions. In most situations, we will be
using the one above, but the latter sheds light on the importance of symmetric groups, and
occasionally comes in handy.
Definition 1.1.3 (group action)
Group Gacts on a set X6=∅if there exists a homomorphism G→Sym(X), where
Sym(X)is the group of permutations of the set X. We define for each g∈Ga
function as follows: fg:X→X;β7→ g(β).
We immediately become informal and say that “Gacts on X” where Gis understood to be
a group, and Xa nonempty set.
Given a set of nelements, what’s the most natural group you can think of that acts on it?
Chances are you are thinking of the symmetric group on nelements, namely Sn. In order
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