Groups & Rings - Concise notes
MATH50005
PDFs
Problem Sheets - Class
Problem Sheets - Unseen
Table of contents
Colour Code - Definition are green in these notes, Consequences are red and Causes are blue
Content from MATH40003 assumed to be known.
Groups
Homomorphisms + Normal Subgroups
Homomorphisms, Isomorphisms and Automorphisms
Group Axioms
G is a group w.r.t a binary operation $\iff \forall g,h,i \in G$
G1 - $gh \in G$ (Closure Axiom)
G2 - $(gh)i = g(hi)$ (Associativity Axiom)
G3 - $\exists e \in G \text{ s.t } \forall g \in G, ge = e = eg$ (Existence of identity)
G4 - $\forall g \in G, \exists g^{-1} \text{ s.t } gg^{-1} = e = g^{-1}g$ (Existence of inverses)
Definition 1.1
A function $f: G \to H$ is a Homomorphism if $\forall a,b \in G$ we have $f(ab) = f(a)f(b)$.
Note $ab$ operation of G, $f(a)f(b)$ operation of H
corollary: $f(e_{G}) = e_{H} \implies f(g^{-1}) = f(g)^{-1}$
Definition 1.4 A function $f: G \to H$ an Isomorphism if $f$ a bijective homomorphism.
We write $f: G \xrightarrow{\sim} H$ or $G \cong H$
Definition 1.6
A function $f$ an isomorphism, $f: G \xrightarrow{\sim} G$ is called an Automorphism Extend this to define $Aut(G)$ as the group of automorphisms of G under composition. Conjugation by an element in G is an automorphism.
Definition
For a homomorphism $f: G \to H$ assosciate:
$Im(f) = f(G) = { f(x) | x \in G }$
$Ker(f) = {x \in G | f(x) = e_{H}}$
Normal subgroups, quotient groups and the isomorphism theorem
Definition 1.11 Normal Subgroups
We say that N is stable under the conjugation by any element in G.
Definition 1.12 Simple Groups
group $G$ simple if $G$ has no normal subgroups aside from ${e}$ and $G$
Define
$gS := {gs \lvert s \in S}$ - Left cosets of S
$Sg := {sg \lvert s \in S}$ - Right cosets of S
Lemma
$H \subset G$ a subgroup. If $gH = Hg \text{ } \forall g \in G$ $\implies H$ a normal subgroup
Lemma
$N \subset G$ a normal subgroup. Then $(g_{1}N)(g_{2}N) = (g_{1}g_{2}N)$
Define - Quotient Group
$N$ a normal subgroup of $G$. $G/N$ the quotient group of $G$ modulo $N$ is the set of all left cosets of N in G. $G/N = {aN | a \in G}$
Lemma
$N$ a normal subgroup of $G$. The set $G/N$ of left cosets of $G$ modulo $N$ is a group under group law $(g_{1}N, g_{2}N) \mapsto (g_{1}g_{2}N)$.
Lemma
If $G$ is a group and $H$ is a normal subgroup of $G$ such that both $H$ and the quotient group $G/H$ are finitely generated, then $G$ is also finitely generated.
Theorem 1.19 - Isomorphism Theorem
Let $f: G \to H$ a homomorphism of groups. Consider the map $gKer(f) \mapsto f(g)$, this map is an isomorphism of groups.
\[G/Ker(f) \xrightarrow{\sim} f(G)\]Group-Theoretic Constructions
Centre of a Group $Z(G)$$= {a \in G \mid ax = xa, \forall x \in G }$.
It is the set of elements in $G$ that commute with all elements in $G$.
$Z(G) = G \iff G$ abelian.Inner Automorphisms $Inn(G)$
The set of automorphisms formed by the conjugations of elements in $G$, forming a subgroup of $Aut(G)$
Commutator
Write $[a,b] = aba^{-1}b^{-1}$ this is the commutator of $a$ and $b$.
Commutator of a Group
$[G, G]$ is the smallest subgroup in $G$ containing the commutators $[a,b] \forall a,b \in G$. It is the subgroup generated by all the commutators.
$G$ abelian $\iff [G, G] ={e_{G}}$
Sending an element $g \in G$ to the conjugation by $g$ is a homomorphism $G \to Aut(G)$ with image $Inn(G)$ and kernel $Z(G)$. Giving isomorphism $G/Z(G) \xrightarrow{\sim} Inn(G)$. using Theorem 1.19
Lemma 1.21
$G$ a group .Then $[G, G]$ a normal subgroup of $G$ and $G/[G,G]$ is abelian.
Proposition 1.22
$N$ a normal subgroup of $G$. Then $G/N$ is abelian if and only if $N$ contains $[G,G]$
Lemma 1.23
Any subgroup of $G$ containing $[G,G]$ is normal
Behaviour of products of groups in the abelian case:
Lemma 1.25
$G$ an abelian group. If orders of $a,b \in G$ finite, then order of $ab$ is finite and divides $lcm(ord(a),ord(b))$.
Definition - Torsion subgroups
The set of elements of $G$ that have finite order is a subgroup of $G$, denoted $G_{\text{tors}}$. If $G = G_{\text{tors}}$ we say G is a torsion abelian group.
Definition - $p$-subgroups of G
$G$ an abelian group, $p$ a prime number.
The subgroup $G{p} = {g \rvert g \in G \text{ s.t } ord(g) = p^{n}}$ is the $p$-primary subgroup of G
If $G = G{p}$ then $G$ is called a $p$-primary torsion abelian group
Generators.
Lemma 1.29 - $I$ a set s.t $\forall i \in I$, we have subgroups $H_{i} \subset G$. Then $H = \cap_{i\in I}H_{i}$ a subgroup of G.
Definition 1.30. - Generated Groups
$G$ a group, $S \subset G$ a set. Intersection of all subgroups of G that contain S is the subgroup of $G$ generated by $S$ denoted $< S >$.
If $G = < S >$ then we say $S$ generates G.
Definition 1.32 - Finitely generated group
$G$ finitely generated if $\exists$ pos. integer $n$ s.t $G$ generated by $n$ elements.
Groups Acting on Sets
Actions, Orbits and Stabilisers
Definition 2.1 Action
$G$ a group, $X$ a set. Let $S(X)$ be the group of bijections $X\to X$ with composition as the group law. An action of $G$ on $X$ is a homomorphism $G \to S(X)$
Associates each $g \in G$ to a bijective map $X \to X$, thought of as permutation of elements of X.
Equivalent to a function $G \times X \to X$, an action $\iff (g_{1}g_{2}(x) = g_{1}(g_{2}(x)) \forall g_{1},g_{2} \in G$ and $x \in X$
Definition 2.3 Faithful actions
an action of $G$ on $X$ is faithful if $G\to S(X)$ is injective
Equivalently, kernel of $G \to S(x)$ is trivial. $g(x) = g \forall x \implies g= e_{G}$
Definition 2.4.1 Orbit of elements
Let $G \times X \to X$ an action of G on a set X. The $G$-orbit of $x\in X$ is $G(x) = {g(x) | g\in G} \subset X$
Definition 2.4.2 Stabiliser of $x$
\[\text{St}_{G}(x) = \{g \in G | g(x) = x\}\subset G\]Theorem 2.6 Orbit-Stabiliser Theorem
$G \times X \to X$ an action of $G$ on $X$. $\forall x\in X$ the map $g \mapsto g(x)$, gives bijection from set of left cosets $G/St(x) \to G(x)$, the orbit of $x$.
If $G$ a finite group $\implies |G(x)| = |G|/|St(x)| \forall x \in X$.
If $X$ a finite set and $X = \cup_{i=1}^{n}G(x_{i})$ is a disjoint union of $G$-orbits, then
where $[G: St(x_{i})]$ is the index of $St(x_{i})$ in G
Applications of the orbit-stabiliser theorem
Theorem 2.7 - (Cayley).
Let $G$ a finite group of order $n \implies S_{n}$ has a subgroup isomorphic to $G$
Theorem 2.8 - (Cauchy)
$G$ a finite group of order $n$ with $p$ a prime factor of $n \implies$ G has an element of order $p$.
Definition 2.9 - $p$-groups - $p$ a prime, finite group $G$ is a $p$-group if order of $G$ is a power of $p$.
Corollary 2.10
$G$ a $p$-group $\iff$ order of every element of G is a power of $p$.
Theorem 2.11.
$G$ a $p$-group, $p$-prime. Then $Z(G) \neq {e_{G}}$
Definition 2.13 - $G \times X \to X$ an action of $G$ on $X$. If $X = G(x)$ ($X$ a $G$-orbit) for some $x \in X$, then we say $G$ acts transitively on $X$.
Definition 2.14
Let $G \times X \to X$ an action of $G$ on $X$. If $x \in X$ s.t $g(x) = x$. We say $x$ a fixed point.
Fix($g$) $\subset X$ - the set of fixed points of $g \in G$
Theorem 2.15 - (Jordan)
Let $G \times X \to X$ a transitive action of a finite group $G$ on a finite set $X$. Then:
$\exists g \in G$ s.t $Fix(g) = \varnothing$
Corollary 2.16 Let $G \times X \to X$ an action of a finite group $G$ on a finite set $X$ Then the number of $G$-orbits in $X$ is $|G|^{-1}\sum_{g\in G}|Fix(g)|.$
Finitely Generated Abelian Groups
Smith Normal form
Definition 3.1 - Smith Normal Form
$A = (a_{ij}) \in \mathbb{Z}$ a $(m\times n)$ matrix in Smith Normal Form if:
$a_{ij} = 0$ if $i \neq j$ (only diagonal terms are non-zero)
$a_{i} = a_{ii}$. For $k \geq 0, a_{i} > 0$ for $i \leq k, a_{i} = 0,$ for $i > k$
$a_{1} \rvert a_{2} \rvert \dots \rvert a_{k}$
Theorem 3.2
Any Matrix of integer coefficients made into Smith Normal form via row/col operations.
Row Operations:
Swap $i^{\text{th}}$ and $j^{\text{th}}$ row
multiply $i^{\text{th}}$ row by $-1$
replace $i^{\text{th}}$ row; $r_{i}$ by $r_{i} + ar_{j},\ i \neq j, \ a\in \mathbb{Z}$
Notation
\[d(A) - \text{gcd of } (a_{ij})\] \[t(A) - \text{smallest non-zero } |a_{ij}|\]Corollary
$d(A) \rvert t(A) \implies d(A) \leq t(A)$
Lemma
Any matrix $A$ of integer coefficients transformed via row/col operations to $B$ s.t $t(B) = d(B) = d(A)$
Classification of finitely generated abelian groups
Definition 3.4- Free abelian group of rank $n$
\(\mathbb{Z}^{n} := \{(a_{1},\dots,a_{n})|a_{i} \in \mathbb{Z}\}\) Lemma
$\mathbb{Z}^{m} \cong \mathbb{Z}^{m} \implies n = m$, shows rank is well defined
Lemma
Any subgroup of $\mathbb{Z}^{n}$ isomorphic to $\mathbb{Z}^{m}$ for $m \leq n$\
Corollary 3.7
$G$ a finitely generated abelian group.
$\implies \exists$ surjective homomorphism
some $n$
\[\text{Ker}(f) \cong \mathbb{Z}^{m}\]Theorem 3.8
Every finitely generated abelian group is isomorphic to a product of finitely many cyclic groups
Corollary - 3.10; Any finite abelian group isomorphic to a product of its $p$-primary torsion subgroups.
Theorem 3.11
Every finitely generated abelian group isomorphic to a product of finitely many infinite cyclic groups and finitely many cyclic groups of prime power order
The number of infinite cyclic factors and the number of cyclic factos of order $p^{r}$, for $p \in \mathbb{P}, r \in \mathbb{N}_{+}$, depends only on the group.
Rings
Basic Theory of Rings
Motivation
Definition 4.1 - Ring
A ring a set $R$ with $2$ binary operations, $+$ and $\times$, satisfying:
$(R, +)$ an abelian group
$\hookrightarrow$ written additively; $0$ an identity element, with $-x$ the inverse of $x$Multiplication is assosciative
$\hookrightarrow \forall a,b,c \in R \implies (a\cdot b)\cdot c = a\cdot(b \cdot c)$$\exists !$ unit element for multiplication ; $1$
Satisfying: $1x = x1 = x\ \forall x \in R$Distributivity
$\hookrightarrow \forall a,b,c \in R; a(b + c) = ab + ac\ , (a+b)c = ac + bc$
$R$ is closed under both $+$ and $\times$
Say $R$ commutatitive if $xy = yx,\ \forall x,y \in R$
Lemma 4.2 - Properties of rings
$\forall x \in R, x0 = 0x = 0$
$\forall x,y \in R \implies (-x)y = x(-y) = -xy$
$R \neq {0} \implies 1 \neq {0}$
Definition 4.3 - Subring
Subset of a ring which is a ring under the same $+, times$ and same $1$ is a subring
Lemma 4.4
$S$ a non-empty subset of ring $R$ Then;
$S$ a subring of $R \iff 1 \in S$ and $\forall a,b \in S; a+b \in S,\ ab \in S,\ -a \in S$
Definition 4.6 - Invertible Elements
$x\in R$ invertible if $\exists y,z \in R$ s.t $xy = 1$ and $zx = 1$
if $y = z$ denote $x^{-1} = y = z$
Definition 4.6.2 - Multiplicative group of $R$
\[R^{\times} = \{x \in R \mid x \text{ invertible} \}\]Definition 4.8 - Division Ring
A ring where all non-zero elements a division ring
A ring in which every nonzero element $a$ has a multiplicative inverse,
Definition 4.8.2 - Field
A commutative division ring a Field.
Homomorphisms, ideals and quotient rings
Definition 4.12 - Homomorphism of Rings
$R,S$ rings. $f: R\to S$ a homomorphism of rings if
$f:(R, +) \to (S, +)$ a homomorphism of abelian groups
$f(xy) = f(x)f(y)$
$f(1{R}) = 1{S}$
A subset $R’$ of $R$ a subring $\iff$ tautological map $R’ \to R$ a homomorphism of rings
Definition 4.16 - Ideal rings
$R$ a ring, $I \subset R$ ideal if:
$I$ a subgroup of $(R, +)$ w.r.t $+$
$\forall x \in I,\ \forall r \in R \implies$
$I$ left ideal if only $rx \in I$
$I$ right ideal if only $xr \in I$
$I$ $2$-sided ideal if $rx \in I$ and $xr \in I$
Mostly consider commutative rings so one condition is often enough.
An ideal ring not equal to the whole ring a proper ideal
Defintion 4.17 - Quotient Ring
$R$ a ring, $I \subset R$ a proper ideal
Quotient abelian group, $R/I$ with multiplication as in $R$ called a quotient ring of $R$ by ideal $I$
Definition 4.18 - Principal ideal
$R$ a commutative ring.
Take $a \in R$, consider $aR = {ax | x\in R}$, this is an ideal in $R$, called the principal ideal with generator $a$
Definition 4.19 - Types of homomorphisms
A bijective homomorphism of rings $f: R \to S$ called an isomorphism of rings
A homomorphism of rings $R\to R$ an endomorphism of rings
An isomorphism of rings $R \rightarrow{\sim} R$ an automorphism of rings
Theorem 4.20 - (Isomorphism Theorem)
Let $f:R \to S$ a homomorphism of rings.
Then subring $f(S)$ of $S$ is isomorphic to quotient ring $R/\text{Ker}(f)$
Integral domains and fields
Definition 1. Zero-divisors
$R$ a ring. non-zero elements $a,b \in R$ are called zero divisors if $ab=0$
Definition 2. Integral Domain
Commutative ring without zero divisors an integral domain
Lemma 1.
$R$ an integral domain. $ab \in R$
\[aR = bR \iff a = br,\ r \in R^{\times}\]Proposition 4.24.
Every field is an integral domain.
Theorem 1.
Every finite integral domain a field
Corollary 4.26.
$n \in \mathbb{N}_{+}$, ring $\mathbb{Z}/n\mathbb{Z}$ an integral domain $\iff n \in \mathbb{P}$
Definition 3. Subfield
subset $K$ of field $\mathbb{F}$ a subfield of $\mathbb{F}$ if $K$ a field with the same addition and multiplication as in $\mathbb{F}$.
Say $\mathbb{F}$ a field extension of $K$
Proposition 4.28
$\forall$ rings $R$, $\exists!$ homemomorphism of rings $\mathbb{Z}\to R$
Lemma 4.29.
$R$ an integral domain. kernel of unique homomorphism $\mathbb{Z}\to R$ either $0-$ideal; ${0}\subset \mathbb{Z}$ or principal ideal $p\mathbb{Z},\ p\in \mathbb{P}$
Definition 4. Characteristic of integral domain
Characteristic of integral domain $R$ is the unique non-negative generator of the kernel of a homomorphism $\mathbb{Z}\to R$; either $0$ or $p \in \mathbb{P}$.
denoted $\text{Char}(R)$
Definition 5.
$k$ a field, $V$ an abelian group with an action of elements of $k$ (scalars) on elements of $V$ (vectors)
Where for $x \in k,\ v\in V,\ xv \in V$
$1v = v$ and $x(yv) = (xy)v, \forall\ x,y\in k,\ v \in V$
$(x+y)v = xv + yv, \forall\ x,y \in k, v \in V$
$x(v+w) = xv + xw, \forall\ x \in k, \forall\ v,w \ in V$
Lemma 4.32.
field extension $\mathbb{F}$ of $k$ is a vector space over $k$
Theorem 2.
$k$ a field.
if $char(k) = 0 \implies k$ has unique subfield isomorphic to $\mathbb{Q}\implies k$ a vector space over $\mathbb{Q}$
if $char(k) = p \in \mathbb{P} \implies k$ contains unique subfield isomorphic to $\mathbb{F}{p} \implies k$ a vector space over $\mathbb{F}{p}$
Corollary 4.34.
Every finite field has $p^{n}$ elements, $p \in \mathbb{P},\ n \in \mathbb{N}_{+}$
More on ideals
Proposition 4.35.
A commutative ring a field $\iff$ only proper ideal is the zero ideal.
Proposition 4.36.
$f:R \to S$ a homomorphism of rings
$J \subset S$ an ideal $\implies f^{-1}(J)$ an ideal of $R$
Proposition 4.37.
$f: R\to S$ surjective homomorphism of rings.
$I \subset R$ an ideal $\implies f(I)$ an ideal of $S$
The maps \(I \mapsto f(I) \qquad J \mapsto f^{-1}(J)\) give a bijection between ideals of $R$ that contain ker$(f)$ and ideals of $S$
Definition 6.
$R$ a commutative ring. We say a proper ideal $I \subset R$ a prime ideal if quotient ring $R/I$ an integral domain.
Proposition 4.39.
$R$ a commutative ring. Proper ideal $I \subset R$ a maximal ideal if quotient ring $R/I$ a field.
Every Maximal ideal a Prime ideal.
Proposition 4.41.
$I \subset R$ a maximal ideal $\iff$ there is no proper ideal $J \subset R$ s.t $I \subset J$ and $I \neq J$
PID and UFD
Polynomial rings
$R$ an integral domain. $R[t]$ the ring of polynomials in $t$ with coefficients in $R$.
\[R[t] = \{ a_{n}t^{n} + a_{n-1}t^{n-1}+\dots+a_{1}t + a_{0}| a_{i} \in R\}\quad n = deg(p(t))\]Proposition 5.1.
$R$ an integral domain $\implies$
$R[t]$ an integral domain. $R[t]^{\times} = R^{\times}$
Proposition 5.2.
$k$ a field
$\forall a(t), b(t) \in k[t],\ b(t) \neq 0$
$\implies \exists! q(t), r(t) \in k[t]$ s.t \(a(t) = q(t)b(t) + r(t)\) $r(t) = 0$ or $deg(r(t)) < deg(b(t))$
Definition 7.
Integral domain $R$ with a function $\phi: R\backslash {0} \to \mathbb{Z}_{\geq 0}$ a Euclidean domain if
$\phi(xy) \geq \phi(x)\quad \forall$ non-zero $x,y \in R$
$\forall a,b \in R,\ \exists q,r \in R$ s.t $a = qb + r$ where $r = 0$ or $\phi(r)< \phi(b)$
Definition 8.
Integral domain $R$ a principal ideal domain (PID) if every ideal of $R$ is principal. i.e of form $aR, a \in R$
Theorem 3.
Any euclidean domain is a PID.
Factorisation in Integral Domains
Definition 9.
$R$ an integral domain.
non-zero $x \in R\backslash R^{\times}$ an irreducible element if $x$ not a product of $2$ elements of $R\backslash R^{\times}$
Lemma 5.7.
$R$ an integral domain.
if $x$ irreducible, $a\in R^{\times} \implies ax$ also irreducible
Definition 10.
An integral domain $R$ a unique factorisation domain (UFD) if every element of $R\backslash R^{\times}$ a product of finitely many irreducibles.
This decomposition is unique up to changing order of factors and multiplication of factors by elements in $R^{\times}$.
Also called factorial rings
Definition 11.
$R$ an integral domain. $a,b \in R$
Say $a \in R$ divides $b\in R$; $a|b$ if $b=ra,\ r \in R$
$a$ properly divides $b$ if $b = ra$ and $r\not\in R^{\times}$
if $b = ra, r \in R^{\times} \implies a$ and $b$ associates
Proposition 5.10.
$R$ a UFD $\implies \not\exists$ infinite sequence of non-zero elements $r_{1},r_{2},\dots$ of $R$ s.t $r_{n+1}$ properly divides $r_{n}\ \forall n \geq 1$
Proposition 5.11.
Let $R$ be a UFD. If $p$ is irreducible and $p|ab \implies p|a$ or $p|b$\
Theorem 4.
$R$ an integral domain. $R$ a UFD $\iff$
There is no infinite sequence $r_{1},r_{2},\dots$ of elements of $R$ such that $r_{n+1}$ properly divides $r_{n}\ \forall n \geq 1$
For every irreducible elements $p \in R$ if $p \rvert ab \implies p\rvert a$ or $p\rvert b$
Proposition 5.14.
Suppose $R$ a PID and $I_{1} \subset I_{2} \subset \dots$ are ideals in $R$ Then for some $n$ we have $I_n = I_n+1 = \dots$.
We say that an ascending chain of ideals stabilises
Proposition 5.16.
Suppose $R$ a PID. $p \in R$ an irreducible element such that $p|ab \implies p|a$ or $p|b$
Theorem 5.
Every PID is a UFD.
Fields
Field extensions
Definition 12.
An extension of fiels $k \subset K$ is called finite if $K$ a finite-dimensional vector space over $k$
$dim_{k}(K)$ = degree of the extension. We write $[K:k] = dim_{k}(K)$
Theorem 6.
$k \subset F$ and $F \subset K$ field extensions. Then $K$ a finite extension of $k \iff$ $F$ a finite extension of $k$ and $K$ a finite extension of $F$
i.e we have $[K:k] = [K:F][F:k]$
Constructing fields from irreducible polynomials
Proposition 6.3.
Let $R$ a PID and let $a\in R, a\neq 0$.
$aR$ maximal $\iff$ $a$ irreducible.
Corollary 6.4.
$R$ a PID and $a \in R$ irreducible then $R/ar$ a field.
Proposition 6.6.
Let $k$ a field. A polynomial $f(t) \in k[t]$ of degree $2$ or $3$ irreducible $\iff$ has no roots in $k$
Proposition 6.8.
$p \neq 2$ prime. Field $\mathbb{F}_{p} = \mathbb{Z}/p\mathbb{Z}$ contains $(p-1)/2 \geq 1$ non-squares.
$\forall a \in \mathbb{F}{p}$ non-square we have $t^{2} - a$ irreducible in $F{p}[t]$
with $F_{p}[t]/(t^2 -a)\mathbb{F}{p}[t]$ a quadratic extension of $\mathbb{F}{p}$
Existence of finite fields
Lemma 6.10
$k$ a field s.t $char(k) = p$, $p$ a prime. $\forall x,y \in k$
$\forall x,y \in k,\ m \in \mathbb{Z}$
Lemma 6.11
$k$ a field $p(t) = (t-\alpha_1)\dots(t-\alpha_n)$ for $\alpha_i \in k, i \in {1,\dots,n}$
Then $\alpha_i \neq \alpha_j$ for $i \neq j \iff p(t), q(t)$ have no common root.
Theorem 7.
Let $p$ prime, $n \in \mathbb{Z}_{+}$