Notes on Algebra: Chapter 0

Algebra: Chapter 0 is one of those books that hit me at just the right time. I knew virtually nothing about the majority of its contents, but had just enough background to gain a lot of (admittedly still half-baked) understanding by reading it. That makes it possibly the single book I’ve learned more from than any other book I can remember.

In the interest of having a convenient way to track all that new information, I’ve made a bunch of notes—mostly brief reminders about notation, terminology, or brute mathematical facts. I hope these are at least somewhat helpful in this form, although I really do recommend reading the book, since these notes are missing any mathematical background I understood before reading, and don’t explain concepts in a way that’d do more than jog my memory. In many cases, they’re not even sufficient to jog my memory, but hopefully act as a decent index to the right section of the book!

The notes here cover approximately the first third of the book, namely the material on groups, with some sections on rings, sets, and a lightly interwoven treatment of categories relevant to all of those. I could have continued further in typing them up, but my understanding of the material in later chapters is so unthorough that notes about it would amount to little more than a mechanical transcription from ink on paper into \(\LaTeX\) on a blog. I don’t think that’s very helpful to anyone, and notating all this took a large enough amount of time as it is. I hope to someday better understand all of that later material, and to share my learnings however I can.

Set Theory and Categories

Naive Set Theory

The disjoint union (\(\sqcup\)) e.g. \[ A \sqcup B \] can be formed with \(A' = \{0\} \times A, B' = \{1\} \times B\). It lets us uniquely mark elements as belonging to each set before taking the regular union

Functions between sets

Indexed sets are properly thought of as functions \[ I \rightarrow A \] where \(a_i\) is drawn from \(A\), and \(I\) is \[\mathbb{N^\ast} = \mathbb{N} \setminus \{0\}\]

Injection (\[ f(a) = f(b) \implies a = b \]) is drawn \[ \hookrightarrow \] and surjection (\[ \mathrm{im} f = B \]) is drawn \[ \twoheadrightarrow \]

\(g\) is a left inverse of \(f\) if \[ g \circ f = \mathrm{id} \] and \(g\) is a right inverse of \(f\) if \[ f \circ g = \mathrm{id}\]

A function \(f\) has a left inverse iff it is injective. \(f\) has a right inverse iff it is surjective

The right inverses of a surjective function are called sections

Take a function \(f : A \rightarrow B\) and a singleton set \(\{b\}\) such that \(b \in B\). Then, the fiber of \(f\) over \(b\) is a function denoted as \(f^{-1}(b) : \{b\} \rightarrow A' \subseteq A\) such that \(\forall a \in A' \enspace f(a) = b\).

Categories

Discrete categories are categories where the only morphisms are identities

Slice categories \(\textsf{C}_S\) are categories where the objects are morphisms in some ambient category \(\textsf{C}\) with a particular target object of \(\textsf{C}\), and the morphisms are commutative diagrams in \(\textsf{C}\). They are special cases of comma categories, in which morphisms are seen as objects in some other category

Coslice categories take morphisms from a fixed object \(A\) in \(\textsf{C}\) to all objects in \(\textsf{C}\)

Morphisms

See my existing list of morphisms for some mnemonics

A category where every morphism is an isomorphism is a groupoid

\(\mathrm{Aut}_\textsf{C}(A)\) is a group for all objects \(A\) in all categories \(\textsf{C}\)

Groups

Definitions

A group is a groupoid with a single object

The trivial group \((G, \cdot) = (\{e\}, \cdot)\) where \(e \cdot e = e\)

The group of invertible \(n \times n\) matrices with real entries is denoted \(\mathrm{GL}_n(\mathbb{R})\)

Commutative groups are called Abelian groups

The order of an element \(|g|\) is the smallest positive \(n\) for which \(g^n = e\). If not finite, \(|g| = \infty\)

If finite, the order of a group \(|G|\) is the number of elements in the set, else \(|G| = \infty\)

If \(g \in G\) and \(|G|\) is finite, then \(|g|\) divides \(|G|\)

Let \(g \in G\) be an element of finite order. Then \[ |g^m| = \frac{\mathrm{lcm}(m, |g|)}{m} = \frac{|g|}{\mathrm{gcd}(m, |g|)} \]

Examples of groups

The symmetric group \(S_A\) is the group of permutations of the set \(A\), denoted \(\mathrm{Aut}_{\textsf{Set}}(A)\)

\(S_n\), the group of permutations of \(\{\textbf{1},...,\textbf{n}\}\), is noncommutative for \(n \geq 3\)

Every product of elements in \(S_3\) reduces to one of \[e, y, y^2, x, xy, xy^2\] where \(x\) maps \((1, 2, 3)\) to \((2, 1, 3)\) and \(y\) maps \((1, 2, 3)\) to \((3, 1, 2)\). Note \(x^2 = e\), \(y^3 = e\), and \(yx = xy^2\)

A subset \(A\) of a group \(G\) generates \(G\) if every element of \(G\) is a product of elements of \(A\) and inverses of elements of \(A\)

The dihedral group \(D_{2n}\) for a regular \(n\)-sided polygon has \(2n\) elements: \(n\) rotations by \(\frac{2\pi}{n}\), and \(n\) reflections about lines through the origin and some vertex (and the midpoint of the opposing side)

\(+\) is the operation of an abelian group on \(\mathbb{Z}/n\mathbb{Z}\) where \([a] + [b] = [a + b]\). These groups \(C_n\) are cyclic groups

\((\mathbb{Z}/n\mathbb{Z}^\ast,\cdot)\) is a group under multiplication

The category \(\textsf{Grp}\)

If \(\varphi : G \rightarrow H\) is a group homomorphism, then \(\varphi(e_G) = \varphi(e_H)\) and \(\forall a, b \in G \enspace \varphi(a \cdot b) = \varphi(a) \cdot \varphi(b)\)

There is a forgetful functor \(\textsf{Grp} \leadsto \textsf{Set}\)

Trivial groups are zero-objects of the category \(\textsf{Grp}\). They’re both initial and final in \(\textsf{Grp}\)

\(G \times H\), the direct product of groups \(G\) and \(H\), is \((g_1, h_1) \cdot (g_2, h_2) = (g_1 g_2, h_1 h_2)\)

The coproduct of \(G\) and \(H\) in \(\textsf{Grp}\) is \(G \ast H\), called the free product

In \(\textsf{Ab}\) (the category with abelian groups as objects) products coincide with coproducts, and \(G \times H\) is called a direct sum (\(G \oplus H\)) when treated as a coproduct

Group homomorphisms

The composition of \(G \rightarrow \{\ast\}\) and \(\{\ast\} \rightarrow H\) is the trivial morphism

A group action of a group \(G\) on an object \(A\) of a category \(\textsf{C}\) is a homomorphism \(G \rightarrow \mathrm{Aut}_{\textsf{C}}(A)\)

\(\mathrm{exp}\) is a homomorphism from group \((\mathbb{R}, +)\) to \((\mathbb{R}^{>0}, \cdot)\)

If \(m \: | \: n\) then there is a homomorphism \(\pi_m^n : \mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/m\mathbb{Z}\) such that \(\pi_m^n([a]_n) = [a]_m\)

A group \(G\) is cyclic if isomorphic to \(\mathbb{Z}\) or to \(C_n = \mathbb{Z}/n\mathbb{Z}\) for some n

If \(p\) is prime, \(((\mathbb{Z}/p\mathbb{Z}^\ast), \cdot)\) is cyclic

Free groups

A free group \(F(A)\) is the group component of an initial object in the category \(\mathscr{F}^A\) whose objects are pairs \((j, G)\) where \(G\) is a group and \(j : A \rightarrow G\) is a set-function from \(A\) to \(G\) with morphisms \(j_1 : A \rightarrow G_1\), \(j_2 : A \rightarrow G_2\), \(\varphi : G_1 \rightarrow G_2\) such that the triangle commutes and \(\varphi\) is a group homomorphism

Free groups arise from treating the underlying set \(A\) as an alphabet (with inverses of each element), constructing words from that alphabet, reducing those words with a reduction \(R\), and for reduced words \(w\) and \(w'\) defining \(w \cdot w' = R(ww')\)

Haskell example

A Cayley graph has vertices corresponding to elements of the group, and edges that connect vertices representing the action of generators

For \(A = \{1, ..., n\}\), \(\mathbb{Z}^{\oplus n}\) is a free abelian group on \(A\)

Subgroups

\((H, \bullet)\) is a subgroup of \((G, \cdot)\) if the inclusion function \(i : H \hookrightarrow G\) is a group homomorphism

Group homomorphisms \(\varphi : G \rightarrow G'\) have a kernel subgroup \(\mathrm{ker} \: \varphi \subseteq G\) and image subgroup \(\mathrm{im} \: \varphi \subseteq G'\)

\(\mathrm{ker} \: \varphi = \varphi^{-1}(e_G) = \{g \in G \: | \: \varphi(g) = e_G\}\)

\(\langle A \rangle\) is the subgroup generated by \(A\) in \(G\). In other words, we have \(\varphi_A : F(A) \rightarrow G\), making a free group from element \(A\) and the image is a subgroup of \(G\)

Group \(G\) is finitely generated if there’s a finite \(A \subseteq G\) such that \(G = \langle A \rangle\). Cyclic groups are finitely generated by singletons

Subgroups of cyclic groups are cyclic groups

Every subgroup of a finitely generated free group is free. Every nontrivial subgroup of \(\mathbb{Z}\) is isomorphic to \(\mathbb{Z}\)

group name definition
\(\mathrm{GL}_n(\mathbb{R})\) general linear group \(n \times n\) invertible matrices with real entries
\(\mathrm{SL}_n(\mathbb{R})\) special linear group \(\{ M \in \mathrm{GL}_n(\mathbb{R}) \: | \: \mathrm{det}(M) = 1 \}\)
\(\mathrm{SL}_n(\mathbb{C})\) special linear group \(\{ M \in \mathrm{GL}_n(\mathbb{C}) \: | \: \mathrm{det}(M) = 1 \}\)
\(\mathrm{O}_n(\mathbb{R})\) orthogonal group \(\{ M \in \mathrm{GL}_n(\mathbb{R}) \: | \: MM^{\mathsf{T}} = M^{\mathsf{T}}M = I_n \}\)
\(\mathrm{SO}_n(\mathbb{R})\) special orthogonal group \(\{ M \in \mathrm{O}_n(\mathbb{R}) \: | \: \mathrm{det}(M) = 1 \}\)
\(\mathrm{U}(n)\) unitary group \(\{ M \in \mathrm{GL}_n(\mathbb{C}) \: | \: MM^{\dagger} = M^{\dagger}M = I_n \}\)
\(\mathrm{SU}(n)\) special unitary group \(\{ M \in \mathrm{U}(n) \: | \: \mathrm{det}(M) = 1 \}\)

Quotient groups

A subgroup \(N\) of \(G\) is normal if \(\forall g \in G, \forall n \in N, gng^{-1} \in N\)

The kernel of a group homomorphism is a normal subgroup

Given \(\pi : G \rightarrow G/H\), \(\mathrm{ker} \: \pi = \{ g \in G \: | \: gH = H \} = H\)

Given groups \(H \subseteq G\) and \(g \in G\), \(gH\) is the left coset of \(H\) in \(G\) with respect to \(g\) and \(Hg\) is the right coset

In group theory, ‘kernel’ and ‘normal subgroup’ are equivalent concepts

Canonical decomposition and Lagrange’s theorem

Just as set-functions are a surjective map followed by a bijective map, followed by an injective map, every group homomorphism \(\varphi : G \rightarrow G'\) can be decomposed as \(G \twoheadrightarrow G/ \: \mathrm{ker} \: \varphi \underset{\widetilde{\varphi}}\rightarrow \mathrm{im} \: \varphi \hookrightarrow G'\)

When \(G \rightarrow G'\) is a surjective group homomorphism, \(G' \cong G / \: \mathrm{ker} \: \varphi\)

A presentation of a group \(G\) is \(G \cong \frac{F(A)}{R} = (A | \mathscr{R})\)

All groups are quotients of free groups, all groups are subgroups of a symmetric group (Cayley’s Theorem)

Let \(H\) be a normal subgroup of a group \(G\), and let \(N\) be a subgroup of \(G\) containing \(H\). Then \(N/H\) is normal in \(G/H\) iff \(N\) is normal in \(G\). If so, \(\frac{G/H}{N/H} \cong \frac{G}{N}\)

Let \(H\), \(K\) be subgroups of \(G\) and assume \(H\) is normal in \(G\). Then \(HK\) is a subgroup of \(G\), \(H\) is normal in \(HK\), \(H \cap K\) is normal in \(K\), and \(\frac{HK}{H} \cong \frac{K}{H \cap K}\)

The index of \(H\) in \(G\), denoted \([G:H]\), is \(|G/H|\)

Lagrange’s theorem: If \(G\) is a finite group and \(H \subseteq G\) is a subgroup, then \(|G| = [G:H]\cdot |H|\). \(|H|\) is a divisor of \(|G|\).

Fermat’s little theorem: When \(p\) is a prime integer and \(a\) is any integer, \(a^p \equiv a \: \mathrm{mod} \: p\)

When \(\varphi : G \rightarrow G\) is a homomorphism of abelian groups, these statements are equivalent:

Group actions

For a group \(G\) acting on objects \(A\) in \(\textsf{Set}\), the action \(\rho : G \times A \rightarrow A\) has \(\forall a \in A \enspace \rho(e_G, a) = a\) and \(\forall g, h \in G, \forall a \in A \enspace \rho(gh, a) = \rho(g, \rho(h, a))\)

A group action of \(G\) on \(A\) in \(\textsf{C}\) is faithful (or effective) if \(\sigma : G \rightarrow \mathrm{Aut}_{\textsf{C}}(A)\) is injective

Cayley’s theorem (redux): Every group acts faithfully on some set. Every group can be realized as a subgroup of a permutation group.

A group action is transitive if \(\forall a, b \in A \: \exists g \in G \enspace b = ga\)

The orbit of \(a \in A\) under an action of group \(G\) is the set \(O_G(a) = \{ ga \: | \: g \in G \}\)

The stabilizer subgroup of \(a\) is the set of elements of \(G\) which fix \(a\), that is \(\mathrm{Stab}_G(a) = \{ g \in G \: | \: ga = a \}\)

Orbits of an action of a group on a set form a partition of that set

For \(O\), the orbit of the action of a finite group \(G\), \(O\) is finite and \(|O|\) divides \(|G|\)

\(|O| \cdot |\mathrm{Stab}_G(a)| = |G|\)

Group objects in categories

We can think of groups as a pair of functions \(m : G \times G \rightarrow G\) and \(\iota : G \rightarrow G\)

Let \(\textsf{C}\) be a category with finite products and a final object \(1\). A group object in \(\textsf{C}\) consists of an object \(G\) of \(C\) and of morphisms \(m : G \times G \rightarrow G\), \(e : 1 \rightarrow G\), \(\iota : G \rightarrow G\) with some diagram commutation requirements

Rings and Modules

Rings

A ring \((R, +, \cdot)\) is an abelian group \((R, +)\) with an extra binary operation \(\cdot\) which is associative and has a two-sided identity. \((R, \cdot)\) is a monoid. The operations distribute such that \((r + s) \cdot t = r \cdot t + s \cdot t\) and \(t \cdot (r + s) = t \cdot r + t \cdot s\)

Some people call rings without identity a rng, some call rings with identity a unital ring

The ring on a trivial group \(\{\ast\}\) with \(\ast \cdot \ast = \ast\) is called the zero-ring, in which \(0 = 1\)

Commutative rings with identity are studied in commutative algebra

An element \(a\) in a ring \(R\) is a left-zero-divisor if \(\exists b =\not 0\) such that \(ab = 0\)

An integral domain is a nonzero commutative ring \(R\) with \(1\) such that \(ab = 0 \implies a = 0 \lor b = 0\)

An element \(u\) of a ring \(R\) is a unit if \(\exists v \in R\) such that \(uv = 1, vu = 1\)

Two-sided units have unique inverses \(u^{-1}\)

A division ring is a ring where every nonzero element is a two-sided unit

A field is a nonzero commutative ring \(R\) with \(1\) where every nonzero element is a unit

If \(R\) is a finite commutative ring, \(R\) is an integral domain iff \(R\) is a field

Wedderburn’s little theorem shows finite division rings are commutative

The group of units in the ring \(\mathbb{Z}/n\mathbb{Z}\) is the group \((\mathbb{Z}/n\mathbb{Z})^\ast\)

\(\mathbb{Z}/p\mathbb{Z} \: \mathrm{integral\:domain} \iff \mathbb{Z}/p\mathbb{Z} \: \mathrm{field} \iff p \: \mathrm{prime}\)

The coefficients of a polynomial are in a ring

The set of polynomials in \(x\) over \(R\) is denoted \(R[x]\)

Rings of power series \(\sum_{i=0}^{\infty} a_ix^i\) are denoted \(R[[x]]\)

Semigroups are sets with associative operations, monoids are semigroups with identity, groups are monoids where each element has an inverse

Given a monoid \((M, \cdot)\) and a ring \(R\) we can create a new ring \(R[M]\) via linear combinations of \(\sum_{m \in M} a_m \cdot m\)

The polynomial ring \(R[x]\) can be interpreted as a monoid ring \(R[\mathbb{N}]\)

There are also group rings, akin to monoid rings but with groups

The category \(\textsf{Ring}\)

For every ring \(R\) there is a group homomorphism \(\varphi : \mathbb{Z} \rightarrow R\) where \((\forall n \in \mathbb{Z}) \enspace \varphi(n) = n \cdot 1_R\)

The kernel of a homomorphism \(\varphi : R \rightarrow S\) of rings is \(\mathrm{ker} \: \varphi = \{ r \in R \: | \: \varphi(r) = 0 \}\)

A subring \(S\) of a ring \(R\) is a ring whose set is a subset and the inclusion function \(S \hookrightarrow R\) is a ring homomorphism

\(\mathrm{End}_\textsf{Ab}(G)\) is a ring for every abelian group \(G\)

Ideals and quotient rings

Let \(R\) be a ring. A subgroup \(I\) of \((R, +)\) is a left ideal of \(R\) if \((\forall r \in R) \enspace rI \subseteq I\). Similarly for right ideals and \(Ir\).

The only ideal of \(R\) containing \(1_R\) is \(R\). Ideals in general are rings without multiplicative identity

The kernel of a ring homomorphism \(\varphi : R \rightarrow S\) is an ideal of \(R\)

The characteristic of a ring \(R\) is \(n\) where \(\mathrm{ker} \: f = n \mathbb{Z}\) and \(f : \mathbb{Z} \rightarrow \mathbb{R}\) is the unique ring homomorphism mapping \(a\) to \(a \cdot 1_R\)

In rings, \(\mathrm{kernel} \iff \mathrm{ideal}\)

Like set functions and group homomorphisms, ring homomorphisms can be decomposed as a surjection to the quotient over the kernel to an image injective to the second ring.

Prime and maximal ideals

A commutative ring \(R\) is Noetherian if every ideal of \(R\) is finitely generated

An integral domain \(R\) is a principal ideal domain (PID) if every ideal of \(R\) is principal

\(\mathbb{Z}\) is a PID

If \(I\) and \(J\) are ideals of \(R\) then \(IJ\) denotes the ideal generated by all products \(ij\) with \(i \in I\), \(j \in J\)

Let \(R\) be a commutative ring, and \(f(x) \in R[x]\) be a monic polynomial of degree \(d\). Then \(\varphi : R[x] \rightarrow R^{\oplus d}\) (defined by sending \(g(x) \in R[x]\) to the remainder of dividing \(g(x)\) by \(f(x)\)) induces an isomorphism of abelian groups \(\frac{R[x]}{(f(x))} \cong R^{\oplus d}\)

\(\frac{\mathbb{R}[x]}{(x^2 + 1)} \cong \mathbb{C}\) as rings

Let \(I =\not (1)\) be an ideal of commutative ring \(R\). \(I\) is a prime ideal if \(R/I\) is an integral domain. \(I\) is a maximal ideal if \(R/I\) is a field.

Let \(I\) be an ideal of a commutative ring \(R\). If \(R/I\) is finite, then \(I\) is prime iff \(I\) is maximal

The set of prime ideals of a commutative ring is called the spectrum, denoted \(\mathrm{Spec}\:R\)

In algebraic geometry, \(\mathbb{C}[x]\) is the ring corresponding to the affine line \(\mathbb{C}\)

The Krull dimension of a commutative ring \(R\) is the length of the longest chain of prime ideals in \(R\)

Modules over a ring

If \(R\) is a ring and \(I \subseteq R\) is a two-sided ideal, then \(R\), \(I\) and \(R/I\) are modules over \(R\)

Freyd-Mitchell embedding theorem: every small abelian category is equivalent to a subcategory of the category of left-modules over a ring

\(R\)-modules are abelian groups with an action of \(R\)

A left-\(R\)-module on an abelian group \(M\) consists of a map \(R \times M \rightarrow M\) mapping \((r, m)\) to \(rm\) such that

Every abelian group is a \(\mathbb{Z}\)-module in exactly one way

\(R\)-modules form a category denoted \(R\text{-}\textsf{Mod}\)

If \(R = k\) is a field, \(R\)-modules are called \(k\)-vector spaces. The category of vector spaces over a field can be denoted \(k\text{-}\textsf{Vect}\). Morphisms in \(k\text{-}\textsf{Vect}\) are called linear maps, and linear algebra is the study of \(k\text{-}\textsf{Vect}\)

The center is the set of elements that commute with all other elements. In a ring, \(ar = ra\) for all \(r\) in \(R\)

Let \(R\) be a commutative ring. An \(R\)-algebra is a ring homomorphism \(\alpha : R \rightarrow S\) such that \(\alpha(R)\) is contained in the center of \(S\)

In the context of \(R\text{-}\textsf{Mod}\), \(\mathrm{kernel} \iff \mathrm{submodule}\)

Products and coproducts in \(R\text{-}\textsf{Mod}\)

\(R[A]\) is a free commutative \(R\)-algebra on the set \(A\)

Noncommutative polynomial rings \(R\langle A \rangle\) are isomorphic to the monoid ring over the free monoid on \(A\), constructed via all finite strings of elements in \(A\) with the concatenation operation

An \(R\)-module \(M\) is Noetherian if every submodule of \(M\) is finitely generated as an \(R\)-module

Following from Hilbert’s basis theorem: If \(R\) is Noetherian as a ring, and \(S\) is a finite-type \(R\)-algebra, then \(S\) is Noetherian

Complexes and homology

A chain complex of \(R\)-modules (or complex) is a sequence of \(R\)-modules and \(R\)-module homomorphisms \[\cdots \overset{d_i+2}\longrightarrow M_{i+1} \overset{d_i+1}\longrightarrow M_{i} \overset{d_i}\longrightarrow M_{i-1} \overset{d_i-1}\longrightarrow \cdots\] such that \((\forall i) \enspace d_i \circ d_{i+1} = 0\)

\((M_\bullet, d_\bullet)\) or simply \(M_\bullet\) denote a complex

When indices increase rather than decrease, we have cochains and cohomology (instead of chains and homology)

The homomorphisms \(d_i\) are called boundary, or differentials

\(d_i \circ d_{i+1} \iff \mathrm{im} \: d_{i+1} \subseteq \mathrm{ker} \: d_i\)

Homology is a measure of the difference between the image of \(d_{i+1}\) and the kernel of \(d_i\), a complex is exact if it has no homology, that is \(\mathrm{im} \: d_{i+1} = \mathrm{ker} \: d_i\)

Trivial modules (denoted \(0\)) have complices exact at that module. A complex is exact (or an exact sequence) if it’s exact at all modules

A complex \(\cdots \longrightarrow 0 \longrightarrow L \overset{\alpha}\longrightarrow M \longrightarrow \cdots\) is exact at \(L\) iff \(\alpha\) is a monomorphism

A short exact sequence runs from \(0\) to \(0\) through injection then surjection

A homomorphism \(\varphi : M \rightarrow M'\) gives rise to the short exact sequence \(0 \longrightarrow \mathrm{ker} \: \varphi \longrightarrow M \longrightarrow \mathrm{im} \: \varphi \longrightarrow 0\)

Split exact sequences are short exact sequences isomorphic to \(0 \longrightarrow M_1 \longrightarrow M_1 \oplus M_2 \longrightarrow M_2 \longrightarrow 0\), which arise from the second projection of a direct sum \(M_1 \oplus M_2 \rightarrow M_2\). We say “the sequence splits”

The \(i\)th homology of a complex is the \(R\)-module \(H_i(M_\bullet) = \frac{\mathrm{ker} \: d_i}{\mathrm{im} \: d_{i+1}}\)

Snake lemma: roughly, we can “snake back” to the beginning of a sequence of homologies to produce a long exact homology sequence, based on multiple short exact sequences linked via homomorphism

Groups, second encounter

The conjugation action

The class formula comes from the action of \(G\) on itself by conjugation

For a group \(G\) acting on a finite set \(S\), with fixed points \(Z = \{ a \in S \: | \: (\forall g \in G) : ga = a \}\) we can count via \[|S| = |Z| + \sum_{a \in A} [G : G_a]\]

A \(p\)-group is a finite group with order a power of a prime \(p\)

The center of a group \(G\) is denoted \(Z(G) = \{ g \in G \: | \: (\forall g \in G) : ga = ag \}\)

Sylow theorems

The Sylow theorems are three statements about \(p\)-subgroups of a finite group \(G\)

Cauchy’s theorem: Let \(G\) be a finite group. If \(p\) is a prime divisor of \(|G|\), then \(G\) contains an element of order \(p\)

A group is simple if it is nontrivial and its only normal subgroups are \(\{e\}\) and the group itself

\(P \subseteq G\) is a \(p\)-Sylow subgroup if it is a \(p\)-group and \(p\) does not divide \([G : P]\)

Sylow theorem 1: Every finite group contains a \(p\)-Sylow subgroup for all primes \(p\)

If \(p^k\) divides the order of \(G\) then \(G\) has a subgroup of order \(p^k\)

Let \(G\) be a finite group, \(P\) be a \(p\)-Sylow subgroup, and \(H \subseteq G\) a \(p\)-group. Then \(H\) is contained in a conjugate of \(P\). \(\exists g \in G\) such that \(H \subseteq gPg^{-1}\)

Sylow theorem 2: Every maximal \(p\)-subgroup in \(|G|\) is a \(p\)-Sylow subgroup, as large as allowed by Langrage’s theorem

Sylow theorem 3: Let \(p\) be a prime integer, and \(G\) be a finite group of order \(|G| = p^rm\). Assume \(p\) does not divide \(m\). Then the number of \(p\)-Sylow subgroups of \(G\) divides \(m\) and is congruent to \(1 \: \mathrm{mod} \: p\)

If \(p < q\) are prime integers and \(q\) is not \(q \: \mathrm{mod} p\) then a group \(G\) of order \(pq\) is cyclic

Let \(q\) be an odd prime, and \(G\) a noncommutative group of order \(2q\). Then \(G \cong D_{2q}\), the dihedral group

Composition series and solvability

Jordan-Hölder theorem: Any two composition series of a group are equivalent

A series of subgroups \(G_i\) of a group \(G\) is a decreasing sequence of subgroups starting from \(G\). \(G = G_0 \subsetneq G_1 \subsetneq G_2 ...\)

The length of a series is the number of strict inclusions

A series is normal if \(G_{i+1}\) is normal in \(G_i\) for all \(i\)

The maximal length of a normal series in \(G\), \(\ell(G)\) measures how far \(G\) is from being simple. \(\ell(G) = 0\) means \(G\) is trivial, and \(\ell(G) = 1\) means \(G\) is simple