MATH 541 (Modern
Algebra I)

Definitions and Theorems

Ruixuan
Tu

ruixuan.tu@wisc.edu

University of Wisconsin-Madison

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

**[Def] injective:**\(f: A\to B\) is injective if \(f(a)=f(b)\implies a=b\)**[Def] surjective:**\(f: A\to B\) is surjective if \(\forall b\in B, \exists a\in A\) s.t. \(f(a)=b\)**[Def] bijective:**\(f: A\to B\) is bijective if both injective and surjective**[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 \(\forall b\in B, f^{-1}\left( f(b) \right)=b\)**[Def] product of sets:**\(A, B\) sets, \(A\times B=\left\{ (a, b) : a\in A, b\in B \right\}\)**[Def] binary operation \(\star\)**on a set \(X\) is a function s.t. \(\star: X\times X\to X\), \((x, y)\mapsto x+y\)**[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. \(a\star a^{-1}=a^{-1}\star a=e\)

**[Def] abelian/commutative group:**if \(\forall a, b\in G, a\star b=b\star a\), then group \(G\) is called abelian/commutative**[Def] \(\operatorname{Aut}\) automorphism (self map)**is an operation/transformation on an object, after which the object coincides with itself. \(\text{Aut}([n])=\left\{ f: [n]\to [n] : f \text{ is a bijection} \right\}\)**[Thm]**\((\operatorname{Aut}, \circ)\) forms a group**[Def] \(S_n\) symmetry group of \(n\) elements**is defined by \(S_n=\operatorname{Aut}([n])\)

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

**[Def] subspace:**for \(W\subseteq V\) with \(V\) a vector space, \(W\) is a subspace of \(V\) \(\iff\) closed on \(+\) and scalar multiplication**[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. \(b=ak\)**[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**[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 \(0\leq r\leq |a|\)**[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. \(\alpha|a\), \(\alpha|b\)**[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. \(\forall s\in S, s\leq s_{\max}\)

**[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 \(n|b-a\)**[Def] \(\bar{a}\)**for \(a\in \mathbb{Z}\), write \(\bar{a}\) for its congruence class**[Def] \(\mathbb{Z}/n \mathbb{Z}\)**is a group with \(+\) defined by \(\bar{a}+\bar{b}=\overline{a+b}\)**[Def] isomorphic**: two groups \(G, H\) are isomorphic if \(\exists \varphi: G\to H\) bijection of sets s.t. \(\forall a, b\in G, \varphi(a) \varphi(b)=\varphi(ab)\) \(\forall a, b\in G\)**[Thm]**\(R=\left\{ 1, r, r^2, \dots \right\} \subseteq D_{2n}\), \(R\cong \mathbb{Z}/n \mathbb{Z}\) isomorphic

**[Def] group morphism/homomorphism:**let \(G, H\) be groups, a morphism \(\varphi : G \to H\) is a map of sets s.t. \(\forall x, y\in G, \varphi(x) \star_{H} \varphi(y)=\varphi(x \star_{G} y)\)**[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 \(\forall \bar{x}\in V, \varphi(c\bar{x})=c\varphi(\bar{x})\)**[Def] faithful action:**an action \(\varphi\) is faithful iff \(\varphi\) is injective**[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)**[Thm] of linear transformation:**\(\varphi(\bar{0})=\bar{0}\)**[Thm] of morphism:**\(\varphi(e_G)=e_H\) for a group morphism \(\varphi: G\to H\); \(\varphi(x^{-1})=\varphi(x)^{-1}\); \(\varphi(x\cdot x^{-1})=\varphi(e_G)=\varphi(x)\varphi(x^{-1})\)**[Rmk]**\(V=\text{vector space}\), \((V, +)=\text{abelian group}\), \(\varphi: \underset{\text{linear transformation}}{V\to W}\) is in particular a morphism, \((V, +)\to (W, +)\)**[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**[Thm]**An automorphism of \(\square\) (\(\operatorname{Aut}(\square)\)) is completely determined by its induced auto on the set of vertices. \(D_8=\operatorname{Aut}(\square)\overset{\varphi}{\to}\operatorname{Aut}([4])=S_4\), \([4]=\text{set of vertices}\)

**[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**[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

**[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**[Def] transposition:**a transposition in \(S_n\) is an element of the form \((a_1, a_2)\) (in cycle representation), for \(a_1, a_2\in [n]\)**[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 \(i\)**[Thm]**a group morphism is injective iff the kernel is \({1}\)**[Def] kernel of group morphism:**\(\ker(\varphi)=\left\{ g\in G: \varphi(g)=1 \right\}\)

**[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 \(g_1=g_2\)**[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 \(g_1=g_2\)**[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. \(f=h\circ g\)**[Thm]**for the \(g, h\) in the definition of factorization, if \(g\) is surjective, then \(h\) has to be unique if it exists**[Def] binary relation:**a binary relation \(\sim\) on \(X\) is a subset \(R\subseteq X\times X\) s.t. \(x\sim y\) iff \((x, y)\in R\)**[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 (\(x\sim y ~\land~ y\sim z\implies x\sim z\))**[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. \(x\in X_i\)**[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. \(x\in X_\pi(G)\)

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

**[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**[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)**[Thm]**if \(H=\ker(\varphi)\) for some \(\varphi: G\to G'\), then \(H\) is automatically a subgroup

**[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 \(\operatorname{Stab}_G(x)=\left\{ g\in G: gx=x \right\}\)**[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)**[Thm]**\(G_x\) is a subgroup

**[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 \([H]\in 2^G\)**[Rmk]**in general, suppose \(G\) acts on \(X\). The kernel of the action \(\ker(G\to \operatorname{Aut}(X))\) is the intersection \(\cap_{x\in X} \operatorname{Stab}_G(x)\)**[Def] normal subgroup:**\(H\leq G\) is called normal if \(N_G(H)=G\)**[Thm]**- kernels of group morphisms are always normal
- stabilizers are not always normal. Note that every subgroup is the stabilizer of some element in a suitable group action

**[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 \(G\)**[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 \(GL_n(\mathbb{Z})\)

**[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)**[Thm] Lagrange’s Thm:**If \(G\) is a finite group and \(H\) is a subgroup of \(G\), then \(|H|\) divides \(|G|\)

**[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**[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)**[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)

**[Def] group structure:**group**[Def] compatible with group structure:**keeps the group law (from Yibo)**[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., \(N_G(K)=G\))**[Def] natural projection:**i.e., homomorphism, a logical way of mapping an algebraic structure onto its quotient structures (from Wolfram MathWorld)**[Def] quotient group**\(H=G/K\) such that if \(\varphi: G\to H = G/K\) is the natural projection \(\varphi(g)\star_H \varphi(g')=\varphi(g\star_G g')\)

**[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 \(G/K\)**[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., \(\varphi'=\phi\circ \varphi\)**[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 \(G/K\)**[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 \(Hg\)**[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 \(|G|/|H|\)**[Cor]**If \(K\trianglelefteq G\), then \(\underset{\text{group formed by left/right cosets}}{|G/K|}=|G|/|K|\). Note that \(gK=Kg\)**[Thm]**If \(G\) is a finite group, \(x\in G\), then \(|x|\) divides \(|G|\)**[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)**[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)- Commutative diagram in Lecture 13+14.pdf

**[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)**[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 \(\rho(H)\leq \rho(H')\)

**[Def] simple group:**A group \(G\) is simple if it has no normal subgroups other than \(G\) or \(\{1\}\)**[Example]**\(\mathbb{Z}/n \mathbb{Z}\) is simple if and only if \(n\) is prime**[Lemma]**Any group of order prime \(p\) is isomorphic to \(\mathbb{Z}/p\mathbb{Z}\)**[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 \(i\)**[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

**[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 \(A_n\)**[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. \(\sigma = \alpha \tau \alpha^{-1}\)**[Prop]**For \(\sigma=(a_1, \dots, a_m)\), \(\operatorname{sgn}(\sigma)=(-1)^{m-1}\)

- cyclic group