MATH 541 (Modern Algebra I)
Definitions and Theorems

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

Lecture 2 (0127) Dihedral Groups

10. [Def] usually, we write $ab$ for $a\star b$, $1$ for $e$ (identity), or $0$ for $e$ (if $G$ is abelian)
11. [Def] order of group $|g|$ of an element $g\in G$ is the smallest $n\in {\mathbb{Z}}_{>0}$ s.t. $g^n=\underset{n\text{ times}}{g\cdots g}=1$, $g^0=1$, $g^{-n}=(g^{-1}{)}^n$
12. [Def] Dihedral group: $D_{2n}=⟨r,s:r^n=1,s^2=1,rs=s{r}^{-1}⟩$, 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 $V$ $⟺$ 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 $\mathrm{\exists }k\in \mathbb{Z}$ 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 $a,b\in \mathbb{Z},a\ne 0$, there exists a unique pair $(q,r)\in {\mathbb{Z}}^2$ s.t. $b=qa+r$ and $0\le r\le |a|$
5. [Def] greatest common divisor: let $a,b\in \mathbb{N}={\mathbb{Z}}_{\ge 0}$, we say that $d\in \mathbb{N}$ is the greatest common divisor of $a,b$, noted by $d=\gcd (a,b)$, if $\alpha \le d$ 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 $\mathrm{\exists }!s_{min}$ in $S$ s.t. $\mathrm{\forall }s\in S,s\ge s_{min}$
   • let $S\subseteq \mathbb{Z}$ which is bounded above, then $\mathrm{\exists }!s_{max}$ in $S$ s.t. $\mathrm{\forall }s\in S,s\le s_{max}$
7. [Def] relatively prime: two numbers $a,b\in \mathbb{N}$ are called relatively prime, if $\gcd (a,b)=1$. We write $a\equiv b\mathrm{mod}n$ $(a,b\in \mathbb{Z},n\in \mathbb{N})$ if $n|b-a$
8. [Def] $\overline{a}$ for $a\in \mathbb{Z}$, write $\overline{a}$ for its congruence class
9. [Def] $\mathbb{Z}/n\mathbb{Z}$ is a group with $+$ defined by $\overline{a}+\overline{b}=\overline{a+b}$
10. [Def] isomorphic: two groups $G,H$ are isomorphic if $\mathrm{\exists }\phi :G\to H$ bijection of sets s.t. $\mathrm{\forall }a,b\in G,\phi (a)\phi (b)=\phi (ab)$ $\mathrm{\forall }a,b\in G$
11. [Thm] $R=\{1,r,r^2,\dots \}\subseteq D_{2n}$, $R\cong \mathbb{Z}/n\mathbb{Z}$ isomorphic

Lecture 4 (0201) Homomorphism

1. [Def] group morphism/homomorphism: let $G,H$ be groups, a morphism $\phi :G\to H$ is a map of sets s.t. $\mathrm{\forall }x,y\in G,\phi (x){\star }_H\phi (y)=\phi (x{\star }_{G}y)$
2. [Def] linear transformation: $V,W$ are vector spaces, then a morphism (aka linear transformation) $\phi :V\to W$ is a map of sets s.t. $\mathrm{\forall }\overline{x},\overline{y}\in V\phi (\overline{x}+\overline{y})=\phi (\overline{x})+\phi (\overline{y})$ and $\mathrm{\forall }\overline{x}\in V,\phi (c\overline{x})=c\phi (\overline{x})$
3. [Def] faithful action: an action $\phi$ is faithful iff $\phi$ 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: $\phi (\overline{0})=\overline{0}$
6. [Thm] of morphism: $\phi (e_G)=e_H$ for a group morphism $\phi :G\to H$; $\phi (x^{-1})=\phi (x{)}^{-1}$; $\phi (x\cdot x^{-1})=\phi (e_G)=\phi (x)\phi (x^{-1})$
7. [Rmk] $V=\text{vector space}$, $(V,+)=\text{abelian group}$, $\phi :\underset{\text{linear transformation}}{V\to W}$ is in particular a morphism, $(V,+)\to (W,+)$
8. [Lemma] Say $\phi :G\hookrightarrow H$ is an injective group morphism. Then $x_1{x}_2\cdots x_n=1$ in $G$ holds iff $\phi (x_1)\cdots \phi (x_n)=\phi (x_1{x}_2\cdots x_n)=1$ in $H$. Notation: $\hookrightarrow$ injective
9. [Thm] An automorphism of $◻$ ($\mathrm{Aut}(◻)$) is completely determined by its induced auto on the set of vertices. $D_8=\mathrm{Aut}(◻)\stackrel{\phi }{\to }\mathrm{Aut}([4])=S_4$, $[4]=\text{set of vertices}$

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: $G\to \mathrm{Aut}(X)\stackrel{def}{=}\{f:X\to X:f\text{ bijection}\}$. 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., $\mathrm{\forall }g\in S_n$, $\mathrm{\exists }$ 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: $\ker (\phi )=\{g\in G:\phi (g)=1\}$

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 $\mathrm{\forall }{g}_1,g_2:C\to A$ 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 $\mathrm{\exists }h:C\to B$ 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 ($x\sim y⟺y\sim x$), and transitive ($x\sim y\wedge y\sim z⟹x\sim z$)
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 $\mathrm{\forall }i,j\in I,i\ne j,x_i\cap x_j=\mathrm{\varnothing }$] $⟺$ $\mathrm{\forall }x\in X,\mathrm{\exists }!i\in I$ s.t. $x\in X_i$
8. [Thm] when the definition of partition is satisfied, we have a well-defined map $X\stackrel{\pi }{\to }I$ s.t. $\mathrm{\forall }x\in X,\pi (x)\in 2^X$ 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, $\mathrm{\forall }a\in \mathbb{Z}$, $a^p\equiv a\mathrm{mod}p$
2. [Thm] given $x,y\in \mathbb{N}$, $d\stackrel{def}{=}\gcd (x,y)$; then $\mathrm{\exists }\lambda ,\mu \in \mathbb{Z}$ s.t. $\lambda x+\mu y=d$
3. [Thm] (Lagrange Thm) if $H$ is a subgroup of a finite group $G$, then $|H|||G|$. in particular, $g\in G$, then $|g|||G|$

Lecture 9 (0213) Subgroups

1. [Def] subgroup: a non-empty subset $H\subseteq G$ with $G$ be a group is called a subgroup (notation: $H\le G$) 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 $H\le G$ (from Dummit)
3. [Thm] if $H=\ker (\phi )$ for some $\phi :G\to G^{\prime }$, 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 ${\mathrm{Stab}}_G(x)=\{g\in G:gx=x\}$
2. [Def] orbit: let $G$ be a group of permutations of a set $S$; for each $x$ in $S$, let ${\mathrm{Orb}}_G(x)=\{\phi (x):\phi \in G\}$. The set ${\mathrm{Orb}}_G(x)$ is a subset of $S$ called the orbit of $x$ under $G$. We use $|{\mathrm{Orb}}_G(x)|$ to denote the number of elements in ${\mathrm{Orb}}_G(x)$ (from Contemporary Abstract Algebra)
3. [Thm] $G_x$ is a subgroup

Lecture 11 (0217) Normalizers and centralizers

1. [Def] normalizer: $N_G(H)=\{g\in G:gH{g}^{-1}=H\}={\mathrm{Stab}}_G([H])$, $G\circlearrowleft G$ (acts on) by conjugation and $[H]\in 2^G$
2. [Rmk] in general, suppose $G$ acts on $X$. The kernel of the action $\ker (G\to \mathrm{Aut}(X))$ is the intersection ${\cap }_{x\in X}{\mathrm{Stab}}_G(x)$
3. [Def] normal subgroup: $H\le G$ 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, $C_G(A)=\{g\in G:ga=ag,\mathrm{\forall }a\in A\}=$ set of all elements that commute with everything in $G$
6. [Thm] The subgroup $\{\left(\begin{array}{cccc}\lambda & & & \\ & \lambda & & \\ & & \ddots & \\ & & & \lambda \end{array}\right):\lambda \ne 0\}\subseteq G{L}_n(\mathbb{Z})$ is the center of $G{L}_n(\mathbb{Z})$

Lecture 12 (0220) Review

1. [Def] conjugation: Let $G$ be a group and let $x\in G$; then $x$ defines a homomorphism ${\phi }_x:G\to G$ given by ${\phi }_x(g)=xg{x}^{-1}$. This is a homomorphism. The operation on $G$ given by ${\phi }_x$ 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 $\phi :G\stackrel{\text{surjective}}{\to }H$ is a homomorphism. Then $H\to 2^G$ $h\mapsto {\phi }^{-1}(h)$ defines a partition of $G$. Alternatively, this partition is also the one induced by the equivalance relation $\sim$ on $G$ induced by $K:=\ker (\phi )$ via $g\sim g^{\prime }⟺g^{\prime }=kg$ for some $k\in K$; the $\sim$ on $G$ can be viewed as given by the action of $\ker (\phi )$ 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 $K\le G$ 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 $\phi :G\to H=G/K$ is the natural projection $\phi (g){\star }_H\phi (g^{\prime })=\phi (g{\star }_G{g}^{\prime })$

Lecture 14 (0227)

1. [Thm] If $K\le G$ is normal, $\mathrm{\exists }$ a surjective group morphism $G\stackrel{\phi }{\to }H$ s.t. $K=\ker (\phi )$. We write this $H$ as $G/K$
2. [Thm] If ${\phi }^{\prime }:G\to H^{\prime }$ is any other group homomorphism such that $K\le \ker ({\phi }^{\prime })$, then $\mathrm{\exists }!\varphi :H\to H^{\prime }$ s.t. $$\begin{array}{ccc}G& \stackrel{\phi }{\to }& H\\ & \underset{{\phi }^{\prime }}{\searrow }& \downarrow \varphi \\ & & H^{\prime }\end{array}$$ commutes, i.e., ${\phi }^{\prime }=\varphi \circ \phi$
3. [Cor] If $K\le G$ is normal (notation $K⊴G$), and $\phi :G\to H$ and ${\phi }^{\prime }:G\to H^{\prime }$ are two surjective group morphisms such that $\ker (\phi )=\ker ({\phi }^{\prime })=K$, then $\mathrm{\exists }!H\to H^{\prime }$ s.t. $$\begin{array}{ccc}G& \stackrel{\phi }{\to }& H\\ & \underset{{\phi }^{\prime }}{\searrow }& \downarrow \\ & & H^{\prime }\end{array}$$ commutes. In this sense we say that the morphisms $\phi$ and ${\phi }^{\prime }$ are canoncially isomorphic. In particular, $H$ and $H^{\prime }$ are canonically isomorphic to $G/K$
4. [Def] coset: For any $H\le G$, 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 $H\le G$, then $|H|$ divides $|G|$ and the number of left/right cosets is equal to $|G|/|H|$
6. [Cor] If $K⊴G$, 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 $\phi :G_1\to G_2$ be a group homomorphism, let $\ker (\phi )$ be the kernel of $\phi$, then $\mathrm{Img}(\phi )\cong G_1/\ker (\phi )$ where $\cong$ denotes group isomorphism (from ProofWiki)
9. [Thm] second isomorphism theorem: Let $G$ be a group, and let $H\le G$, $N⊴G$, 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 $H,N⊴G$, $N\subseteq H$, then $N⊴H$, $H/N⊴G/N$, $\frac{G/N}{H/N}\cong \frac{G}{H}$ (from ProofWiki)
2. [Thm] fourth isomorphism theorem: Let $G$ be a group, $N⊴G$, then $A$ be (the set of subgroups $H\le G$ s.t. $N\le H$) $\to$ (the set of subgroups of $G/N$), i.e., $H\mapsto H/N$ is a bijection denoted by $\rho$. Moreover, if $H\le H^{\prime }$ in $A$, then $\rho (H)\le \rho (H^{\prime })$

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 $\{1\}⊴G_r⊴\cdots ⊴G_1⊴G_0=G$ 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 $\{1\}⊴G_r⊴\cdots ⊴G_1⊴G_0=G$ and $\{1\}⊴G_s^{\prime }⊴\cdots ⊴G_1^{\prime }⊴G_0^{\prime }=G$, then $r=$ and $\mathrm{\exists }\sigma \in S_r$ (symmetry group of a set of $r$ elements) s.t. $G_i/G_{i+1}\cong G_{\sigma (i)}^{\prime }/G_{\sigma (i+1)}^{\prime }$ 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 $\mathrm{sgn}(\pi )$ 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 $\mathrm{sgn}:S_n\to C_2$ 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 $\sigma =(a_1,\dots ,a_m),\tau =(b_1,\dots ,b_m),\mathrm{\exists }\alpha \in S_n$ s.t. $\sigma =\alpha \tau {\alpha }^{-1}$
3. [Prop] For $\sigma =(a_1,\dots ,a_m)$, $\mathrm{sgn}(\sigma )=(-1{)}^{m-1}$

Lecture 18 (0308)

Lecture 19 (0310)

Lecture 20 (0320)

1. cyclic group