MATH 541 (Modern Algebra I) Definitions and Theorems

Ruixuan Tu
ruixuan.tu@wisc.edu

Preliminaries

1. $$a$$ divides $$b$$: $$a|b$$, $$ak=b$$ for some $$k\in \mathbb{Z}$$
2. monomorphism (injective) $$\hookrightarrow$$
3. epimorphism (surjective) $$\twoheadrightarrow$$

Lecture 1 (0125) Intro to Groups

1. [Def] injective: $$f: A\to B$$ is injective if $$f(a)=f(b)\implies a=b$$
2. [Def] surjective: $$f: A\to B$$ is surjective if $$\forall b\in B, \exists a\in A$$ s.t. $$f(a)=b$$
3. [Def] bijective: $$f: A\to B$$ is bijective if both injective and surjective
4. [Def] $$f$$ is bijective $$\iff$$ f has a unique inverse $$f^{-1}$$ s.t. $$\forall a\in A, f^{-1}\left( f(a) \right)=a$$ and $$\forall b\in B, f^{-1}\left( f(b) \right)=b$$
5. [Def] product of sets: $$A, B$$ sets, $$A\times B=\left\{ (a, b) : a\in A, b\in B \right\}$$
6. [Def] binary operation $$\star$$ on a set $$X$$ is a function s.t. $$\star: X\times X\to X$$, $$(x, y)\mapsto x+y$$
7. [Def] group $$G$$ is a set equipped with a binary operation $$\star$$ s.t.
• (associativity) $$\forall a, b, c\in G, (a\star b)\star c=a\star (b\star c)$$
• (identity element $$e$$) $$\exists e\in G$$ s.t. $$\forall a\in G, e\star a=a\star e=a$$
• (inverse element) $$\forall a\in G, \exists a^{-1}\in G$$ s.t. $$a\star a^{-1}=a^{-1}\star a=e$$
8. [Def] abelian/commutative group: if $$\forall a, b\in G, a\star b=b\star a$$, then group $$G$$ is called abelian/commutative
9. [Def] $$\operatorname{Aut}$$ automorphism (self map) is an operation/transformation on an object, after which the object coincides with itself. $$\text{Aut}([n])=\left\{ f: [n]\to [n] : f \text{ is a bijection} \right\}$$
10. [Thm] $$(\operatorname{Aut}, \circ)$$ forms a group
11. [Def] $$S_n$$ symmetry group of $$n$$ elements is defined by $$S_n=\operatorname{Aut}([n])$$

Lecture 2 (0127) Dihedral Groups

1. [Def] usually, we write $$ab$$ for $$a\star b$$, $$1$$ for $$e$$ (identity), or $$0$$ for $$e$$ (if $$G$$ is abelian)
2. [Def] order of group $$|g|$$ of an element $$g\in G$$ is the smallest $$n\in \mathbb{Z}_{> 0}$$ s.t. $$g^n=\underset{n\text{ times}}{g\cdots g}=1$$, $$g^0=1$$, $$g^{-n}=(g^{-1})^n$$
3. [Def] Dihedral group: $$D_{2n}= \left< r, s : r^n=1, s^2=1, rs=sr^{-1} \right>$$, number of elements $$\left| D_{2n} \right|=2n$$

Lecture 3 (0130) Modular Arithmetic

1. [Def] subspace: for $$W\subseteq V$$ with $$V$$ a vector space, $$W$$ is a subspace of $$V$$ $$\iff$$ closed on $$+$$ and scalar multiplication
2. [Def] division: let $$a, b\in \mathbb{Z}$$, say $$a$$ divides $$b$$ / $$b$$ is divisible by $$a$$, noted by $$a|b$$, if $$\exists k\in \mathbb{Z}$$ s.t. $$b=ak$$
3. [Def] prime number: say $$p\in \mathbb{Z}_{>1}$$ is a prime number if the only $$a\in \mathbb{Z}_{>1}$$ that divides $$p$$ is $$p$$ itself
4. [Thm] for any $$a, b\in \mathbb{Z}, a\not= 0$$, there exists a unique pair $$(q, r)\in \mathbb{Z}^2$$ s.t. $$b=qa+r$$ and $$0\leq r\leq |a|$$
5. [Def] greatest common divisor: let $$a,b \in \mathbb{N}=\mathbb{Z}_{\geq 0}$$, we say that $$d\in \mathbb{N}$$ is the greatest common divisor of $$a,b$$, noted by $$d=\operatorname{gcd}(a,b)$$, if $$\alpha \leq d$$ for every $$\alpha \in \mathbb{N}$$ s.t. $$\alpha|a$$, $$\alpha|b$$
6. [Def] well-ordering principle
• let $$S\subseteq \mathbb{Z}$$ which is bounded below, then $$\exists! s_{\min}$$ in $$S$$ s.t. $$\forall s\in S, s\geq s_{\min}$$
• let $$S\subseteq \mathbb{Z}$$ which is bounded above, then $$\exists! s_{\max}$$ in $$S$$ s.t. $$\forall s\in S, s\leq s_{\max}$$
7. [Def] relatively prime: two numbers $$a, b\in \mathbb{N}$$ are called relatively prime, if $$\operatorname{gcd}(a, b)=1$$. We write $$a\equiv b \mod n$$ $$(a, b\in \mathbb{Z}, n\in \mathbb{N})$$ if $$n|b-a$$
8. [Def] $$\bar{a}$$ for $$a\in \mathbb{Z}$$, write $$\bar{a}$$ for its congruence class
9. [Def] $$\mathbb{Z}/n \mathbb{Z}$$ is a group with $$+$$ defined by $$\bar{a}+\bar{b}=\overline{a+b}$$
10. [Def] isomorphic: two groups $$G, H$$ are isomorphic if $$\exists \varphi: G\to H$$ bijection of sets s.t. $$\forall a, b\in G, \varphi(a) \varphi(b)=\varphi(ab)$$ $$\forall a, b\in G$$
11. [Thm] $$R=\left\{ 1, r, r^2, \dots \right\} \subseteq D_{2n}$$, $$R\cong \mathbb{Z}/n \mathbb{Z}$$ isomorphic

Lecture 4 (0201) Homomorphism

1. [Def] group morphism/homomorphism: let $$G, H$$ be groups, a morphism $$\varphi : G \to H$$ is a map of sets s.t. $$\forall x, y\in G, \varphi(x) \star_{H} \varphi(y)=\varphi(x \star_{G} y)$$
2. [Def] linear transformation: $$V, W$$ are vector spaces, then a morphism (aka linear transformation) $$\varphi: V\to W$$ is a map of sets s.t. $$\forall \bar{x}, \bar{y}\in V \varphi(\bar{x}+\bar{y})=\varphi(\bar{x})+\varphi(\bar{y})$$ and $$\forall \bar{x}\in V, \varphi(c\bar{x})=c\varphi(\bar{x})$$
3. [Def] faithful action: an action $$\varphi$$ is faithful iff $$\varphi$$ is injective
4. [Def] faithful action: an action or operation of $$G$$ is said to be faithful if $$K={e}$$; that is, the kernel of $$G\to \pi(S)$$ is trivial (from Serge Lang Algebra)
5. [Thm] of linear transformation: $$\varphi(\bar{0})=\bar{0}$$
6. [Thm] of morphism: $$\varphi(e_G)=e_H$$ for a group morphism $$\varphi: G\to H$$; $$\varphi(x^{-1})=\varphi(x)^{-1}$$; $$\varphi(x\cdot x^{-1})=\varphi(e_G)=\varphi(x)\varphi(x^{-1})$$
7. [Rmk] $$V=\text{vector space}$$, $$(V, +)=\text{abelian group}$$, $$\varphi: \underset{\text{linear transformation}}{V\to W}$$ is in particular a morphism, $$(V, +)\to (W, +)$$
8. [Lemma] Say $$\varphi: G\hookrightarrow H$$ is an injective group morphism. Then $$x_1x_2\cdots x_n=1$$ in $$G$$ holds iff $$\varphi(x_1)\cdots \varphi(x_n)=\varphi(x_1x_2\cdots x_n)=1$$ in $$H$$. Notation: $$\hookrightarrow$$ injective
9. [Thm] An automorphism of $$\square$$ ($$\operatorname{Aut}(\square)$$) is completely determined by its induced auto on the set of vertices. $$D_8=\operatorname{Aut}(\square)\overset{\varphi}{\to}\operatorname{Aut}([4])=S_4$$, $$[4]=\text{set of vertices}$$

Lecture 5 (0203) Homomorphism (continued)

1. [Example] group homomorphism: $$G$$ is group, $$X$$ is set, an action of $$G$$ on $$X$$ is a group homomorphism: $$G\to \operatorname{Aut}(X)\overset{def}{=} \left\{ f: X\to X : f \text{ bijection} \right\}$$. group law = composition
2. [Rmk] homomorphism: In general if $$G$$ is given by generators and relations, to check whether $$\psi: G\to H$$ ($$H$$ is group) is a group morphism, it suffices to check that $$\psi$$ represents the relations

Lecture 6 (0206) Symmetry groups and cubes

1. [Thm] symmetry group of cube: we have 9+8+6+1=24 elements in the symmetry group $$G$$ of a cube. It turns out that every element in $$G$$ can be obtained by composing Type III rotations (rotations about an axis through the center of the cube, sloped), so that the number of transpositions in $$S_4$$ is 6
2. [Def] transposition: a transposition in $$S_n$$ is an element of the form $$(a_1, a_2)$$ (in cycle representation), for $$a_1, a_2\in [n]$$
3. [Thm] $$S_n$$ is generated by transpositions, i.e., $$\forall g\in S_n$$, $$\exists$$ an expression $$g=g_1 \cdot \cdots \cdot g_m$$ for some $$m$$ s.t. $$g_i$$ is a transposition for every $$i$$
4. [Thm] a group morphism is injective iff the kernel is $${1}$$
5. [Def] kernel of group morphism: $$\ker(\varphi)=\left\{ g\in G: \varphi(g)=1 \right\}$$

Lecture 7 (0208) Abstract Nonsense

1. [Thm] say $$f: A\to B$$ is a map of sets, then $$f$$ is surjective iff it has the following property: for every $$g_1, g_2: B\to C$$ (C is some other set) s.t. $$g_1\circ f= g_2\circ f$$, then $$g_1=g_2$$
2. [Thm] $$f: A\to B$$ is injective iff $$\forall g_1, g_2: C\to A$$ s.t. $$f\circ g_1 = f\circ g_2$$, then $$g_1=g_2$$
3. [Def] factorization: let $$f: A\to B, g: A\to C$$ be maps of sets, we say that $$f$$ factors through $$g$$ if $$\exists h: C\to B$$ s.t. $$f=h\circ g$$
4. [Thm] for the $$g, h$$ in the definition of factorization, if $$g$$ is surjective, then $$h$$ has to be unique if it exists
5. [Def] binary relation: a binary relation $$\sim$$ on $$X$$ is a subset $$R\subseteq X\times X$$ s.t. $$x\sim y$$ iff $$(x, y)\in R$$
6. [Def] equivalance relation: we say that $$\sim$$ is an equivalance relation if it is reflexive ($$x\sim x$$), symmetric ($$x\sim y\iff y\sim x$$), and transitive ($$x\sim y ~\land~ y\sim z\implies x\sim z$$)
7. [Def] partition: a partition of a set $$X$$ is given by a subset $$I\subseteq 2^{X}$$ s.t. [$$\cup_{i\in I} X_i=X$$ and $$\forall i, j\in I, i\not= j, x_i\cap x_j=\emptyset$$] $$\iff$$ $$\forall x\in X, \exists! i\in I$$ s.t. $$x\in X_i$$
8. [Thm] when the definition of partition is satisfied, we have a well-defined map $$X\overset{\pi}{\to}I$$ s.t. $$\forall x\in X, \pi(x)\in 2^X$$ is the unique element s.t. $$x\in X_\pi(G)$$

Lecture 8 (0210) Abstract Nonsense (continued)

1. [Thm] (Fermat’s Little Thm) let p be a prime number, $$\forall a\in \mathbb{Z}$$, $$a^p\equiv a \mod{p}$$
2. [Thm] given $$x, y\in \mathbb{N}$$, $$d\overset{def}{=}\operatorname{gcd}(x, y)$$; then $$\exists \lambda, \mu\in \mathbb{Z}$$ s.t. $$\lambda x + \mu y=d$$
3. [Thm] (Lagrange Thm) if $$H$$ is a subgroup of a finite group $$G$$, then $$|H| \large| \normalsize |G|$$. in particular, $$g\in G$$, then $$|g| \large| \normalsize |G|$$

Lecture 9 (0213) Subgroups

1. [Def] subgroup: a non-empty subset $$H\subseteq G$$ with $$G$$ be a group is called a subgroup (notation: $$H\leq G$$) if it is closed under inversion and group multiplication; in particular, $$H$$ is itself a group
2. [Def] subgroup: Let $$G$$ be a group. The subset $$H$$ of $$G$$ is a subgroup of $$G$$ if $$H$$ is nonempty and $$H$$ is closed under products and inverses (i.e., $$x, y\in H$$ implies $$x^{-1}\in H$$ and $$xy\in H$$). If $$H$$ is a subgroup of $$G$$ we shall write $$H\leq G$$ (from Dummit)
3. [Thm] if $$H=\ker(\varphi)$$ for some $$\varphi: G\to G'$$, then $$H$$ is automatically a subgroup

Lecture 10 (0215) Stabilizers

1. [Def] stabilizer: let $$G\circlearrowleft X$$ (acts on) with $$X$$ be a set, take $$x\in X$$, then the stabilizer of $$x$$ in $$G$$, also denoted by $$G_x$$ is $$\operatorname{Stab}_G(x)=\left\{ g\in G: gx=x \right\}$$
2. [Def] orbit: let $$G$$ be a group of permutations of a set $$S$$; for each $$x$$ in $$S$$, let $$\operatorname{Orb}_G(x)=\left\{ \varphi(x): \varphi\in G \right\}$$. The set $$\operatorname{Orb}_G(x)$$ is a subset of $$S$$ called the orbit of $$x$$ under $$G$$. We use $$\left\vert \operatorname{Orb}_G(x) \right\vert$$ to denote the number of elements in $$\operatorname{Orb}_G(x)$$ (from Contemporary Abstract Algebra)
3. [Thm] $$G_x$$ is a subgroup

Lecture 11 (0217) Normalizers and centralizers

1. [Def] normalizer: $$N_G(H)=\left\{ g\in G: g H g^{-1} = H \right\}=\operatorname{Stab}_G\left( [H] \right)$$, $$G\circlearrowleft G$$ (acts on) by conjugation and $$[H]\in 2^G$$
2. [Rmk] in general, suppose $$G$$ acts on $$X$$. The kernel of the action $$\ker(G\to \operatorname{Aut}(X))$$ is the intersection $$\cap_{x\in X} \operatorname{Stab}_G(x)$$
3. [Def] normal subgroup: $$H\leq G$$ is called normal if $$N_G(H)=G$$
4. [Thm]
1. kernels of group morphisms are always normal
2. stabilizers are not always normal. Note that every subgroup is the stabilizer of some element in a suitable group action
5. [Def] centeralizer: let $$A\subseteq G$$ be any subset, $$C_G(A)=\left\{ g\in G: ga=ag, \forall a\in A \right\}=$$ set of all elements that commute with everything in $$G$$
6. [Thm] The subgroup $$\left\{ \begin{pmatrix} \lambda & & & \\ & \lambda & & \\ & & \ddots & \\ & & & \lambda \end{pmatrix} : \lambda \not=0 \right\}\subseteq GL_n(\mathbb{Z})$$ is the center of $$GL_n(\mathbb{Z})$$

Lecture 12 (0220) Review

1. [Def] conjugation: Let $$G$$ be a group and let $$x\in G$$; then $$x$$ defines a homomorphism $$\varphi_x: G\to G$$ given by $$\varphi_x(g)=x g x^{-1}$$. This is a homomorphism. The operation on $$G$$ given by $$\varphi_x$$ is called conjugation by $$x$$. (from Wolfram MathWorld)
2. [Thm] Lagrange’s Thm: If $$G$$ is a finite group and $$H$$ is a subgroup of $$G$$, then $$|H|$$ divides $$|G|$$

Lecture 13 (0224) Quotient Groups

1. [Thm] Suppose that $$\varphi: G\overset{\text{surjective}}{\to} H$$ is a homomorphism. Then $$H\to 2^G$$ $$h\mapsto \varphi^{-1}(h)$$ defines a partition of $$G$$. Alternatively, this partition is also the one induced by the equivalance relation $$\sim$$ on $$G$$ induced by $$K:= \ker(\varphi)$$ via $$g\sim g' \iff g'=kg$$ for some $$k\in K$$; the $$\sim$$ on $$G$$ can be viewed as given by the action of $$\ker(\varphi)$$ on $$G$$ by left multiplication
2. [Def] natural map/canonical map: a map or morphism between objects that arises naturally from the definition or the construction of the objects (from Wikipedia)
1. [Example] If $$N$$ is a normal subgroup of a group $$G$$, then there is a canonical surjective group homomorphism from $$G$$ to the quotient group $$G/N$$, that sends an element $$g$$ to the coset determined by $$g$$. (from Wikipedia)
3. [Def] group structure: group
4. [Def] compatible with group structure: keeps the group law (from Yibo)
5. [Thm] Consider $$K\leq G$$ any subgroup defines a partition $$H$$ on $$G$$ such that the partition $$H\subseteq 2^G$$ has a group structure compatible with $$G$$ (i.e., the natural map $$G\to H$$, here is a surjective group morphism). If such a group structure exists, then it is unique and $$K$$ becomes the kernel of $$G\to H$$. We have seen that a necessary condition is that $$K$$ is normal (i.e., $$N_G(K)=G$$)
6. [Def] natural projection: i.e., homomorphism, a logical way of mapping an algebraic structure onto its quotient structures (from Wolfram MathWorld)
7. [Def] quotient group $$H=G/K$$ such that if $$\varphi: G\to H = G/K$$ is the natural projection $$\varphi(g)\star_H \varphi(g')=\varphi(g\star_G g')$$

Lecture 14 (0227)

1. [Thm] If $$K\leq G$$ is normal, $$\exists$$ a surjective group morphism $$G\overset{\varphi}{\to} H$$ s.t. $$K=\ker(\varphi)$$. We write this $$H$$ as $$G/K$$
2. [Thm] If $$\varphi': G\to H'$$ is any other group homomorphism such that $$K\leq \ker(\varphi')$$, then $$\exists! \phi: H\to H'$$ s.t. $\begin{array}{ccc} G & \xrightarrow{\varphi} & H \\ & \underset{\varphi'}{\searrow} & \downarrow \phi \\ & & H' \end{array}$ commutes, i.e., $$\varphi'=\phi\circ \varphi$$
3. [Cor] If $$K\leq G$$ is normal (notation $$K\trianglelefteq G$$), and $$\varphi: G\to H$$ and $$\varphi': G\to H'$$ are two surjective group morphisms such that $$\ker(\varphi)=\ker(\varphi')=K$$, then $$\exists! H\to H'$$ s.t. $\begin{array}{ccc} G & \xrightarrow{\varphi} & H \\ & \underset{\varphi'}{\searrow} & \downarrow \\ & & H' \end{array}$ commutes. In this sense we say that the morphisms $$\varphi$$ and $$\varphi'$$ are canoncially isomorphic. In particular, $$H$$ and $$H'$$ are canonically isomorphic to $$G/K$$
4. [Def] coset: For any $$H\leq G$$, a left coset of $$H$$ is a subset of $$G$$ of the form $$gH$$ for some $$g\in G$$. Similarly, a right coset is of the form $$Hg$$
5. [Thm] Lagrange’s Thm: If $$G$$ is finite and $$H\leq G$$, then $$|H|$$ divides $$|G|$$ and the number of left/right cosets is equal to $$|G|/|H|$$
6. [Cor] If $$K\trianglelefteq G$$, then $$\underset{\text{group formed by left/right cosets}}{|G/K|}=|G|/|K|$$. Note that $$gK=Kg$$
7. [Thm] If $$G$$ is a finite group, $$x\in G$$, then $$|x|$$ divides $$|G|$$
8. [Thm] first isomorphism theorem: Let $$\varphi: G_1\to G_2$$ be a group homomorphism, let $$\ker(\varphi)$$ be the kernel of $$\varphi$$, then $$\operatorname{Img}(\varphi)\cong G_1/\ker(\varphi)$$ where $$\cong$$ denotes group isomorphism (from ProofWiki)
9. [Thm] second isomorphism theorem: Let $$G$$ be a group, and let $$H\leq G$$, $$N\trianglelefteq G$$, then $$\frac{H}{H\cap N}\cong \frac{HN}{N}$$ (from ProofWiki)
10. Commutative diagram in Lecture 13+14.pdf

Lecture 15 (0301)

1. [Thm] third isomorphism theorem: Let $$G$$ be a group, and let $$H, N\trianglelefteq G$$, $$N\subseteq H$$, then $$N\trianglelefteq H$$, $$H/N \trianglelefteq G/N$$, $$\frac{G/N}{H/N}\cong \frac{G}{H}$$ (from ProofWiki)
2. [Thm] fourth isomorphism theorem: Let $$G$$ be a group, $$N\trianglelefteq G$$, then $$A$$ be (the set of subgroups $$H\leq G$$ s.t. $$N\leq H$$) $$\to$$ (the set of subgroups of $$G/N$$), i.e., $$H\mapsto H/N$$ is a bijection denoted by $$\rho$$. Moreover, if $$H\leq H'$$ in $$A$$, then $$\rho(H)\leq \rho(H')$$

Lecture 16 (0303)

1. [Def] simple group: A group $$G$$ is simple if it has no normal subgroups other than $$G$$ or $$\{1\}$$
2. [Example] $$\mathbb{Z}/n \mathbb{Z}$$ is simple if and only if $$n$$ is prime
3. [Lemma] Any group of order prime $$p$$ is isomorphic to $$\mathbb{Z}/p\mathbb{Z}$$
4. [Def] composition series: Let $$G$$ be a finite group. A composition series is a sequence $$\{1\}\trianglelefteq G_r\trianglelefteq \cdots \trianglelefteq G_1\trianglelefteq G_0 = G$$ such that $$G_i/G_{i+1}$$ is simple for every $$i$$
5. [Thm] Jordan-Hölder Theorem: Let $$G$$ be a finite group. Suppose that $$\{1\} \trianglelefteq G_r \trianglelefteq \cdots \trianglelefteq G_1 \trianglelefteq G_0 = G$$ and $$\{1\} \trianglelefteq G'_s \trianglelefteq \cdots \trianglelefteq G'_1 \trianglelefteq G'_0 = G$$, then $$r=$$ and $$\exists \sigma\in S_r$$ (symmetry group of a set of $$r$$ elements) s.t. $$G_i/G_{i+1} \cong G'_{\sigma(i)}/G'_{\sigma(i+1)}$$ for every $$i$$. This is analogous to prime factorization of integers

Lecture 17 (0306)

1. [Def] alternating group: Let $$S_n$$ denote the symmetric group on $$n$$ elements. For any $$\pi \in S_n$$, let $$\operatorname{sgn}(\pi)$$ be the sign of $$\pi$$ (i.e., $$\mapsto \{\pm 1\}=\mathbb{Z} / 2 \mathbb{Z}$$). With $$C_2= \mathbb{Z} / 2 \mathbb{Z}$$, the kernel of the mapping $$\operatorname{sgn}: S_n\to C_2$$ is called the alternating group on $$n$$ elements $$A_n$$
2. [Lemma] Any two cycles of the same lengths in $$S_n$$ are conjugate. That is, for any $$\sigma=(a_1, \dots, a_m), \tau=(b_1, \dots, b_m), \exists \alpha\in S_n$$ s.t. $$\sigma = \alpha \tau \alpha^{-1}$$
3. [Prop] For $$\sigma=(a_1, \dots, a_m)$$, $$\operatorname{sgn}(\sigma)=(-1)^{m-1}$$

Lecture 20 (0320)

1. cyclic group