Reading Notes

MATH 403

Table of contents

Chapter 7: Groups
Chapter 8: Normal Subgroups and Quotient Groups
Chapter 9: Topics in Group Theory
1. 9.1: Direct Products

Textbook: Abstract Algebra, an Introduction. Thomas Hungerford. Internet Archive link

Chapter 7: Groups

Algebraic systems like $\mathbb{Z}$ and $\mathbb{Z}_n$ have two operations
Groups have a single operation

7.1: Definition and Examples of Groups

The integers form a group under addition but not multiplication
Nonzero rational numbers form a group under multiplication but not addition
Study of permutations: most historically important
A permutation is an ordering of elements in a set $T$
A permutation is a bijective function from $T$ to $T$
Every bijection has an inverse function

A group is a nonempty set $G$ equipped with a binary operation $*$ that satisfies the following axioms:

Closure: $a*b \in G$ for all $a,b \in G$
Associativity: $(a*b)*c = a*(b*c)$ for all $a,b,c \in G$
Identity: there exists an element $e \in G$ such that $a*e = e*a = a$ for all $a \in G$
Inverses: for each $a \in G$ there exists an element $a^{-1} \in G$ such that $a*a^{-1} = a^{-1}*a = e$

A group is abelian if $a*b = b*a$ for all $a,b \in G$ (i.e. commutativity).

The order of a group is the number of elements it has.
$S_n$: the symmetric grup on $n$ symbols with order $n!$.
$A(T)$ is the set of all permutations of $T$ (all bijective functions $T \to T$. $A(T)$ is a group under the operation of composition of functions.
Dihedral group / group of symmetries of the square.

Groups and Rings

Theorem 7.1. Every ring is an abelian group under addition. A nonzero ring $R$ is never a group under multiplication.

Theorem 7.2. The nonzero elements of a field $F$ form an abelian group under multiplicaiton.

A ring $R$ with identity always has at least one subset that is a group under multiplication

Theorem 7.3. If $R$ is a ring with identity, then the set $U$ of all units in $R$ is a group under multiplication.

Theorem 7.4. Let $G$ with an operation $*$ and $H$ with operation $\diamond$ be groups. Define an operation $\square$ on $G \times H$ by

\[(g, h) \square (g', h') = (g * g', h \diamond h')\]

Then $G \times H$ is a group under $\square$. If $G$ and $H$ are abelian, then $G \times H$ is abelian. If $G$ and $H$ are finite, then $G \times H$ is finite and $|G \times H| = |G| \cdot |H|$.

7.2: Basic Properties of Groups

Standard multiplicative notation: write $ab$ instead of $a * b$.

Theorem 7.5. Let $G$ be a group and let $a, b, c \in G$. Then

$G$ has a unique identity element
Cancelation holds in $G$: if $ab = ac$, then $b = c$; if $ba = ca$, then $b = c$
Each element of $G$ has a unique inverse.

Corollary 7.6. If $G$ is a gruop and $a, b \in G$, then

Order of inverse application: $(ab)^{-1} = b^{-1}a^{-1}$
Canceling of inverses: $(a^{-1})^{-1} = a$

Theorem 7.7. Let $G$ be a group and let $a \in G$. Then for all $m, n \in \mathbb{Z}$, $a^m a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$.

Element $a$ has finite order if $a^k = e$ for some positive integer $k$.
The order of $a$ is the smallest positive integer $k$ such that $a^k = e$.
Order of $a$ denoted $\vert a \vert$
Element has infinite order if $a^k \neq e$ for all positive integers $k$.

Theorem 7.8. Let $G$ be a group and let $a \in G$.

If $a$ has infinite order, then the elements $a^k$, $k \in \mathbb{Z}$, are all distinct.
If $a^i = a^j$ with $i \neq j$, then $a$ has finite order.

Theorem 7.9. Let $G$ be a group and $a \in G$ an element of finite order $n$. Then:

$a^k = e$ iff $n \vert k$
$a^i = a^j$ iff $i \equiv_n j$
If $n = td$, with $d \ge 1$, then $a^t$ has order $d$

Corollary 7.10. Let $G$ be an abelian group in which every element has finite order. If $c \in G$ is an element of largest order in $G$, then the order of every element of $G$ divides $\vert c \vert$.

7.3: Subgroups

A subset of $H$ of a group $G$ is a subgroup of $G$ if $H$ is itself a group under the operation of $G$.

The trivial subgroup of $G$ is $\{e\}$.

It is never necessary to check associativity for a subset $H$ of a group $G$ to be a subgroup of $G$.

Theorem 7.11. A nonempty subset $H$ of a group $G$ is a subgroup of $G$ iff

if $a, b \in H$, then $ab \in H$
if $a \in H$, then $a^{-1} \in H$

Theorem 7.12. Let $H$ be a nonempty finite subset of a group $G$. If $H$ is closed under the operation in $G$, then $H$ is a subgroup of $G$.

The center of $G$ is $Z(G) = \{a \in G \vert ag = ga \text{ for all } g \in G\}$. An element of $G$ is in $Z(G)$ iff it commutes with every element of $G$.

Theorem 7.13. The center of a group $G$ is a subgroup of $G$.

Theorem 7.14. If $G$ is a group and $a \in G$, then $\langle a \rangle = \{ a^n \vert n \in \mathbb{Z} \}$ is a subgroup of $G$.

$\langle a \rangle$ is the cyclic subgroup generated by $a$. If $\langle a \rangle = G$, then $G$ is cyclic. Every cyclic group is abelian.

Theorem 7.15. Let $G$ be a gruop and let $a \in G$.

If $a$ has infinite order, then $\langle a \rangle$ is an infinite subgroup consisting of the distinct elements $a^k, k \in \mathbb{Z}$.
If $a$ has finite order $n$, then $\langle a \rangle = \{e, a, a^2, \dots, a^{n-1}\}$ is a finite subgroup of $G$ with order $n$.

Theorem 7.15 (Additive Version). Let $G$ be an additive group and let $a \in G$.

If $a$ has infinite order, then $\langle a \rangle$ is an infinite subgroup consisting of the distinct elements $ka, k \in \mathbb{Z}$.
If $a$ has finite order $n$, then $\langle a \rangle = \{0, a, 2a, \dots, (n-1)a\}$ is a finite subgroup of $G$ with order $n$.

Theorem 7.16. Let $F$ be any one of $\mathbb{Q}, \mathbb{R}, \mathbb{C}, or \mathbb{Z}_p$, where $p$ is prime. ($F$ is a field.) Then $F^* = F \setminus \{0\}$ be the multiplicative group of nonzero elements of $F$. If $G$ is a finite subgroup of $F^*$, then $G$ is cyclic.

Theorem 7.17. Every subgroup of a cyclic group is itself cyclic.

Theorem 7.18. Let $S$ be a nonempty subset of a group $G$. Let $\langle S \rangle$ be the set of all possible products, in every order, of elements of $S$ and their inverses. Then

$\langle S \rangle$ is a subgroup of $G$ that contains the set $S$
If $H$ is a subgroup of $G$ that contains the set $S$, then $H$ contains the entire subgroup $\langle S \rangle$.

Shows that $\langle S \rangle$ is the smallest subgroup of $G$ containing $S$.
In the case where $S = \{a\}$, $\langle S \rangle = \langle a \rangle$.
$S$ generates $G$

7.4: Isomorphisms and Homomorphisms

Isomorphic groups have the same structure

Definition of an isomorphism. Let $G, H$ be gruops with the group operation $*$. $G$ is isomorphic to $H$ if there is a function $\phi: G \to H$ such that

$\phi$ is a bijection ($\phi$ is injective and surjective)
$\phi(a * b) = \phi(a) * \phi(b)$ for all $a, b \in G$

If $G$ is abelian and $H$ is nonabelian, then $G$ is not isomorphic to $H$.
If $\phi$ is an isomorphism, then $a$ and $\phi(a)$ have the same order
$\phi: G \to G$ is an automorphism if $\phi$ is an isomorphism from $G$ to $G$.

Theorem 7.19. Let $G$ be a cyclic group.

If $G$ is infinite, then $G$ is isomorphic to $\mathbb{Z}$.
If $G$ is finite of order $n$, then $G$ is isomorphic to $\mathbb{Z}_n$.

Definition of a homomorphism. Let $G, H$ be groups with the group operation $*$. A homomorphism from $G$ to $H$ is a function $\phi: G \to H$ such that $\phi(a * b) = \phi(a) * \phi(b)$ for all $a, b \in G$.

Theorem 7.20. Let $G, H$ be groups with identity elements $e_G, e_H$. If $\phi: G \to H$ is a homomorphism, then

\[\phi(e_G) = e_H\]
$\phi(a^{-1}) = \phi(a)^{-1}$ for all $a \in G$
$\text{Im}(f)$ is a subgroup of $H$
If $f$ is injective, $G \cong \text{Im}(f)$

Theorem 7.21. Cayley’s Theorem. Every group $G$ is isomorphic to a group of permutations.

Corollary 7.22. Every finite group $G$ of order $n$ is isomorphic to a subgroup of $S_n$.

A homomorphism from $G$ to a group of permutation is a representation of $G$. Group theory can be reduced to the study of permutation groups.

7.5: The Symmetric and Alternating Groups

Cycle notation
Two cycles are disjoint if they have no elements in common

Theorem 7.23. If $\sigma = (a_1 a_2 \hdots a_k)$ and $\tau = (b_1 b_2 \hdots b_r)$ are disjoint cycles in $S_n$, then $\sigma \tau = \tau \sigma$.

Theorem 7.24. Every permutation in $S_n$ is a product of disjoint cycles.

Theorem 7.25. The order of a permutation $\tau$ in $S_n$ is the LCM of the lengths of the disjoint cycles whose product is $\tau$.

A 2-cycle is a transposition.
Every transposition is its own inverse.
If $\sigma_1 \sigma_2 \sigma_3 \hdots \sigma_{n}$ are transpositions, then the inverse is $\sigma_{n} \sigma_{n-1} \hdots \sigma_2 \sigma_1$.
Note that $(1) = (12)(12)$, $(123) = (12)(23)$, and $(1234) = (12)(23)(34)$

Theorem 7.26. Every permutation in $S_n$ is a product of (not necessarily disjoint) transpositions.

A permutation is even if it is a product of an even number of transpositions.
A permutation is odd if it is a product of an odd number of transpositions.
No permutation is both even and odd.

Lemma 7.27. The identity permutation in $S_n$ is even and not odd.

Theorem 7.28. No permutation in $S_n$ is both even and odd.

The set of all even permutations in $S_n$ is the alternating group $A_n$.

Theorem 7.29. $A_n$ is a subgroup of $S_n$ of order $n! / 2$.

Chapter 8: Normal Subgroups and Quotient Groups

8.1: Congruence and Lagrange’s Theorem

In the integers, $a \equiv b \mod n$ iff $n \vert (a - b)$, or $a - b \in n\mathbb{Z}$.

Definition. Let $K$ be a subgroup of a group $G$ and let $a, b \in G$. Then $a$ is congruent to $b$ modulo $K$, written $a \equiv b \mod K$, if $a^{-1}b \in K$.

Theorem 8.1. Let $K$ be a subgroup of a group $G$. Then the relation of congruence modulo $K$ is

reflexive: $a \equiv a \mod K$
symmetric: if $a \equiv b \mod K$, then $b \equiv a \mod K$
transitive: if $a \equiv b \mod K$ and $b \equiv c \mod K$, then $a \equiv c \mod K$

If $K$ is a subgruop of a group $G$ and if $a \in G$, the congruence class of $a$ mod $K$ is the set of all elemetns of $G$ congruent to $a$ modulo $K$>

\[\{ b \in G \vert b \equiv_K a \} = \{ b \in G \vert ba^{-1} \in K \} = \{b \in G \vert b^{-1} = k, k \in K \}\]

Noting that $b = ka$, we get

\[\{ b \in G \vert b = ka, k \in K \} = \{ ka \vert k \in K \} = Ka \text{ (right coset)}\]

Right coset also denoted $K + a$

Theorem 8.2. Let $K$ be a subgroup of a group $G$ and let $a, c \in G$. Then $a \equiv_K c$ iff $Ka = Kc$.

Corollary 8.3. Let $K$ be a subgroup of a group $G$. Then two right cosets of $K$ are either disjoint or identical.

Lagrange’s Theorem

Theorem 8.4. Let $K$ be a subgroup of a group $G$. Then

$G$ is the union of the right cosets of $K$: $G = \bigcup_{a \in G} Ka$
For each $a \in G$, there is a bijection $f : K \to Ka$. Consequently, if $K$ is finite, any two right cosets of $K$ contain the same number of elements.

If $H$ is a subgroup of a group $G$, the number of distinct right cosets of $H$ in $G$ is the index of $H$ in $G$ and denoted $[G : H]$.
If $G$ is a finite group, then $[G:H]$ is finite

Theorem 8.5. Lagrange’s Theorem If $K$ is a subgroup of a finite group $G$, then the order of $K$ divides the order of $G$. In particular, $\vert G \vert = \vert K \vert [ G : K]$.

Shows us there are limited possibilities for subgroups of a finite group
A subgroup of a group of order 12 must have order 1, 2, 3, 4, 6, or 12

Corollary 8.6. Let $G$ be a finite group.

If $a \in G$, then the order of $a$ divides the order of $G$.
If $\vert G \vert = k$, then $a^k = e$ for every $a \in G$

The Structure of Finite Groups

Major goal of group theory is to classify all finite groups up to isomorphism

Theorem 8.7. Let $p$ be a positive prime integer. Then every group of order $p$ is cyclic and isomorphic to $\mathbb{Z}_p$.

Theorem 8.8. Every group of order 4 is isomorphic to either $\mathbb{Z}_4$ or $\mathbb{Z}_2 \times \mathbb{Z}_2$.

Theorem 8.9. Every group of order 6 is isomorphic to either $\mathbb{Z}_6$ or $S_3$.

order	isomorphic to
1	$\{e\}$
2	$\mathbb{Z}_2$
3	$\mathbb{Z}_3$
4	$\mathbb{Z}_4$ or $\mathbb{Z}_2 \times \mathbb{Z}_2$
5	$\mathbb{Z}_5$
6	$\mathbb{Z}_6$ or $S_3$
7	$\mathbb{Z}_7$

8.2: Normal Subgroups

Suppose $G$ group and $K$$ subgroup.
Create a new group whose elements are right cosets of $K$ in $G$.
Left coset $aK$: $\{ak \vert k \in K\}$
a subgroup $N$ of a group $G$ is normal if $Na = aN$ for every $a \in G$.
Every subgroup of an abelian group is normal.
Note that $Na = aN$ does not imply that $na = na, \forall n \in N$

Theorem 8.10. Let $N$ be a normal subgroup of a group $G$. If $a \equiv_N b$ and $c \equiv_N d$< then $ac \equiv_N bd$.

Theorem 8.11. The following conditions on a subgroup $N$ of a group $G$ are equivalent:

$N$ is normal in $G$
$a^{-1} N a \subseteq N$ for every $a \in G$, where $a^{-1} N a = \{a^{-1}na \vert n \in N\}$
$aNa^{-1} \subseteq N$ for every $a \in G$, where $aNa^{-1} = \{ana^{-1} \vert n \in N\}$
$a^{-1} N a = N$ for every $a \in G$
$aNa^{-1} = N$ for every $a \in G$

8.3: Quotient Groups

$G / N$ denotes the set of all right cosets of $N$ in $G$.

Theorem 8.12. Let $N$ be a normal subgroup of a group $G$. If $Na = Nc$ and $Nb = Nd$ in $G / N$, then $N ab = N cd$.

Theorem 8.13. Let $N$ be a normal subgroup of a group $G$. Then

$G / N$ is a group under the operation defined by $(Na)(Nc) = Nac$.
If $G$ is finite, then the order of $G / N$ is $[G : N]$.
If $G$ is abelian, then $G / N$ is abelian.

If $K$ is the cyclic subgroup $\langle n \rangle$ of $\mathbb{Z}$, then $\mathbb{Z} / K = Z_n$.

8.4: Quotient Groups and Homomorphisms

Definition. Let $\phi: G \to H$ be a homomorphism from a group $G$ to a group $H$. Then the kernel of $\phi$ is the set of all elements of $G$ that are mapped to the identity of $H$.

Kernels are normal subgroups

Theorem 8.16. Let $f : G \to H$ be a homomorphism of groups with kernel $K$. Then $K$ is a normal subgroup of $G$.

The kernel of a homomorphism $f$ measures how far $f$ is from being injective.

Theorem 8.17. Let $f : G \to H$ be a homomorphism of groups with kernel $K$. Then $K = \langle e_G \rangle$ iff $f$ is injective.

Theorem 8.18. If $N$ is a normal subgroup of a group $G$, then the map $\pi G \to G / N$ given by $\pi(a) = Na$ is a surjective homomorphism with kernal $N$.

8.16. states that every kernel is a normal subgroup, but every normal subgroup is also a kernel

Lemma 8.19. Let $: G \to H$ be a group homomorphism with kernel $K$. Let $a, b \in G$. Then $f(a) = f(b)$ iff $Ka = Kb$.

Theorem 8.20. First Isomorphism Theorem. Let $f : G \to H$ be a surjective homomorphism of groups with kernel $K$. Then the quotient group $G / K$ is isomorphic to $H$.

From the first isomorphism theorem, we have a lot of interesting results:

Let $N = \{ a + bi \vert a^2 + b^2 = 1 \}$. Then $\mathbb{C} / N \cong \mathbb{R}^+$.
Let $K = \{1, -1\}$. Then $\mathbb{R}^* / K \cong \mathbb{R}^+$.

Theorem 8.21. Let $N$ be a normal subgroup of a group $G$ and let $K$ be any subgroup of $G$ that contains $N$. Then $K / N$ is a subgroup of $G / N$.

Theorem 8.22. Third Isomorphism Theorem Let $K, N$ be normal subgroups sof a group $G$ with $N \subseteq K \subseteq G$. Then $K / N$ is a normal subgroup of $G / N$, and the quotient group $(G / N) / (K / N)$ is isomorphic to $G / K$.

Corollary 8.23. Let $N$ be a normal subgroup of a group $G$ and let $K$ be any subgroup of $G$ that contains $N$. Then $K$ is normal in $G$ iff $K / N$ is normal in $G / N$.

Theorem 8.24. If $T$ is any subgroup of $G / N$, then $T = H/N$, where $H$ is a subgroup of $G$ that contains $N$.

Classification of finite groups: every finite group is isomorphic to one group on the list
A group is simple if its only normal subgroups are $\langle e \rangle$ and $G$ itself

Theorem 8.25. $G$ is a simple abelian group iff $G$ is isomorphic to the additive group $\mathbb{Z}_p$ for some prime $p$.

Nonabelian simple groups are rare: only five of order les than 1k and 56 of order less than 1m.
Alternating groups: large class of nonabelian simple groups.
Simple groups are the basic building blocks for all groups
If $G$ is a finite group, it only has finitely many normal subgroups other than itself.
Let $G_1$ be a normal subrgroup that has the largest possible order: $G / G_1$ is simple.
- Suppose $G / G_1$ had a proper normal subgroup; it would have form $M / G_1$, where $M$ is a proper normal subgroup of $G$ properly containing $G_1$. But then $M$ would be the largest normal subgroup of $G$, a contradiction.
- If $G_1 \neq \langle e \rangle$, let $G_2$ be a normal subgroup of $G_1$ of largest possible order. We know $G_1 / G_2$ is simple. And we can continue until we reach some $G_n$ that is the identity subgroup, so you get a sequence of ;groups $\langle e \rangle = G_n \subset G_{n-1} \subset \hdots \subset G_3 \subset G_2 \subset G_1 \subset G_0 = G$
- Each $G_i$ is a normal subgroup of its predecessor; each quotient group $G_i / G_{i + 1}$ is simple. These simple groups are the composition factors of $G$
- Composition factors of a finite group $G$ are independent of the choice of the subgroups $G_i$: you can choose different $G_i$ but the simple quotient groups would be isomorphic
- The composition factors of $G$ totally determine the structure of $G$
Classification problem strategy: classify all simple groups and show how the composition factors of an arbitrary group determine the structure of the group

8.5: The Simplicity of $A_n$

Theorem 8.26. For each $n \neq 4$, the alternating group $A_n$ is simple.

Lemma 8.27. Every element of AA_n$with$n \ge 3$$ is a product of 3-cycles.

Lemma 8.28. If $N$ is a normal subgroup of $A_n$ with $n \ge 3$ and $N$ contains a 3-cycle, then $N = A_n$.

Corollary 8.29. If $n \ge 5$, then $(1)$, $A_n$, and $S_n$ are the only normal subgroups of $S_n$.

Chapter 9: Topics in Group Theory

Deeper look into the various aspects of the classification problem of finite gruops.

9.1: Direct Products

if $G, H$ are groups, $G \times H$ is also a group
$G_1 \times G_2 \times \hdots \times G_n$ is the direct product of the groups $G_1, G_2, \hdots, G_n$

Theorem 9.1. let $N_1, N_2, \hdots, N_k$ be normal subgroups of a group $G$ such that every element in $G$ can be written uniquely in the form $a_1 a_2 \hdots a_k$, with $a_i \in N_i$. THen $G$ is isomorphic to the direct product $N_1 \times N_2 \times \hdots \times N_k$.

Lemma 9.2. Let $M, N$ be normal subgroups of a group $G$ such that $M \cap N = \{e\}$. If $a \in M$ and $b \in N$, then $ab = ba$.

Each $N_i$ is a direct factor of $G = N_1 \times \hdots \times N_k$.
$G$ may be the external direct product of the $N_i$ (as tuples) or the internal direct product of the $N_i$ (as elements in the group).

Theorem 9.3. If $M, N$ are normal subgroups of a group $G$ such that $G = MN$ and $M \cap N = \langle e \rangle$, then $G = M \times N$.

order	isomorphic to
1	\(\{e\}\)
2	\(\mathbb{Z}_2\)
3	\(\mathbb{Z}_3\)
4	\(\mathbb{Z}_4\) or \(\mathbb{Z}_2 \times \mathbb{Z}_2\)
5	\(\mathbb{Z}_5\)
6	\(\mathbb{Z}_6\) or \(S_3\)
7	\(\mathbb{Z}_7\)