# Reading Notes

MATH 403

## Table of contents

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

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 \(K\)>

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 \(G = M \times N\).