Search & glossary

Every rule and concept in Groups and Fields, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (79)

$F[x]$ has unique factorization: Every non-constant polynomial over a field is a product of irreducible polynomials, uniquely up to order and constant factors.; Polynomials over a Field › Unique Factorization and the gcd
Abelian groups: A group is abelian (after Niels Abel) if its operation is commutative: $a*b=b*a$ for all $a,b\in G$ .; Groups and the Group Axioms › The Group Axioms
Algebraic elements and the minimal polynomial: An element $\alpha$ of an extension of $F$ is algebraic over $F$ if it is a root of some nonzero polynomial in $F[x]$ .; Field Extensions › Algebraic Elements and Minimal Polynomials
Automorphism: An automorphism of $G$ is an isomorphism $\varphi: G \to G$ from $G$ to itself.; Homomorphisms and Isomorphisms › Automorphisms
Binary operation and closure: A binary operation on a set $S$ is a rule $*$ that takes any ordered pair $a,b\in S$ and returns exactly one value $a*b$ , and that value lies in $S$ again.; Groups and the Group Axioms › Binary Operations
Burnside's lemma: If a finite group $G$ acts on a finite set $X$ , the number of orbits equals $ $\frac{1}{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|,\qquad \operatorname{Fix}(g)=\{x\in X:g\cdot x=x\}.$ $ That is: the number of orbits is the average, over the group, of how many p…; Group Actions › Burnside's Lemma
Cayley's theorem: Every group $G$ is isomorphic to a subgroup of a symmetric group: a subgroup of $\operatorname{Sym}(G)$ , the group of all bijections $G \to G$ under composition, and so a subgroup of $S_n$ when $|G| = n$ is finite.; Permutation Groups › Cayley's Theorem
Center and commutator subgroup: The center $Z(G) = \{z \in G : zg = gz \text{ for all } g \in G\}$ is the set of elements commuting with everything.; Normal Subgroups and Quotient Groups › The Center and the Commutator Subgroup
Composing permutations: For $\sigma, \tau \in S_n$ , the product $\sigma\tau$ is the permutation $(\sigma\tau)(i) = \sigma(\tau(i))$ : apply $\tau$ first, then $\sigma$ , reading right to left like function composition.; Permutation Groups › Composing Permutations
Conjugate elements, conjugacy classes, and the centralizer: For $a, b \in G$ , $b$ is a conjugate of $a$ if $b = gag^{-1}$ for some $g \in G$ .; Conjugacy and the Class Equation › Conjugacy Classes
Conjugate permutations have the same cycle type — and vice versa: For $\sigma,\tau \in S_n$ , $\sigma\tau\sigma^{-1}$ is obtained from $\tau$ 's disjoint cycle decomposition by applying $\sigma$ to every entry: if $\tau = (a_1\,\cdots\,a_k)(b_1\,\cdots\,b_l)\cdots$ , then $στσ^-1 = (σ(a_1) … σ…; Conjugacy and the Class Equation › Conjugacy in Symmetric Groups
Counting by partitions: If $n = p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ , the number of abelian groups of order $n$ , up to isomorphism, is $ $p(a_1)\,p(a_2)\cdots p(a_k),$ $where$ p(a) $is the number of integer partitions of$ a $— the number of ways to write$ a$ as a sum of positive int…; Direct Products and Finite Abelian Groups › Counting Abelian Groups of a Given Order
Cyclic groups and generators: For $a \in G$ , the set of all integer powers $\langle a \rangle = \{a^n : n \in \mathbb Z\}$ (written $\{na : n \in \mathbb Z\}$ when the operation is addition) is always a subgroup of $G$ , the cyclic subgroup generated by $a$ .; Subgroups and Cyclic Groups › Cyclic Groups and Generators
Cyclotomic polynomials: The $n$ -th roots of unity $e^{2\pi ik/n}$ form a cyclic group of order $n$ ; its generators are the primitive $n$ -th roots.; Splitting Fields and Constructions › Roots of Unity and Cyclotomic Polynomials
Degrees multiply in towers: If $F\subseteq E\subseteq K$ are fields with finite degrees, then $[K:F]=[K:E]\,[E:F]$ .; Field Extensions › The Tower Law
Euler's theorem and Fermat's little theorem: For $n \ge 1$ and $\gcd(a,n)=1$ : $a^{\varphi(n)} \equiv 1 \pmod n$ , where $\varphi(n) = |U(n)|$ is Euler's totient.; Cosets and Lagrange's Theorem › Fermat and Euler from Lagrange
Even and odd permutations: A transposition is a $2$ -cycle, swapping two points and fixing the rest.; Permutation Groups › Even and Odd Permutations
Every $p$ -group has a nontrivial center: If $|G|=p^n$ for a prime $p$ and $n\ge1$ , then $Z(G) \ne \{e\}$ ; in fact $p$ divides $|Z(G)|$ .; Conjugacy and the Class Equation › p-Groups and Their Centers
Every finite field has $p^n$ elements: A finite field has prime characteristic $p$ , contains $\mathbb Z_p$ , and is a vector space of some dimension $n$ over it — so it has exactly $p^n$ elements.; Finite Fields › The Size of a Finite Field
Every subgroup of a cyclic group, and how many there are: Every subgroup of a cyclic group is itself cyclic (proved in the final section).; Subgroups and Cyclic Groups › The Subgroups of a Cyclic Group
Extensions and the quotient $F[x]/(p(x))$: A field $E$ containing $F$ is an extension of $F$ , written $E/F$ .; Field Extensions › Building Fields from Polynomials
Field automorphisms fixing a base field: An automorphism of a field $E$ is an isomorphism $\sigma:E\to E$ .; Galois Theory › Automorphisms of Fields
First Isomorphism Theorem: For a homomorphism $\varphi: G \to H$ , $\ker\varphi \trianglelefteq G$ and $G/\ker\varphi \cong \operatorname{im}\varphi$ , via the map $a\,\ker\varphi \mapsto \varphi(a)$ .; Normal Subgroups and Quotient Groups › The First Isomorphism Theorem
Five quick theorems: From the four axioms alone one can prove: the identity $e$ is unique; each $a$ has only one inverse $a^{-1}$ ; $(a^{-1})^{-1}=a$ ; $(ab)^{-1}=b^{-1}a^{-1}$ ; and the cancellation laws — $ax=ay\Rightarrow x=y$ and $xa=ya\Rightarrow x=y$ .; Groups and the Group Axioms › First Consequences of the Axioms
Galois's criterion: A polynomial over $\mathbb Q$ can be solved by radicals (its roots expressed using $+,-,\times,\div$ and $n$ -th roots) exactly when its Galois group is a solvable group.; Galois Theory › Solvability by Radicals
Gauss and Wantzel: A regular $n$ -gon is constructible exactly when $\cos\frac{2\pi}n$ is constructible, which happens exactly when $\varphi(n)$ is a power of $2$ — equivalently, when $n$ is a power of $2$ times a product of DISTINCT Fermat primes $3,5,17,257,65537$ .; Splitting Fields and Constructions › Constructible Regular Polygons
Group action: A (left) group action of $G$ on a set $X$ is a function $G\times X\to X$ , written $(g,x)\mapsto g\cdot x$ , satisfying $e\cdot x=x$ and $(gh)\cdot x=g\cdot(h\cdot x)$ for all $g,h\in G$ and $x\in X$ .; Group Actions › Group Actions
Groups of order $pq$: Let $p<q$ be primes.; The Sylow Theorems › Classifying Groups of Small Order
Homomorphism: A function $\varphi: G \to H$ between groups is a homomorphism if $\varphi(ab) = \varphi(a)\varphi(b)$ for all $a, b \in G$ , where the left-hand product uses $G$ 's operation and the right-hand product uses $H$ 's.; Homomorphisms and Isomorphisms › Homomorphisms
How to find them: To find the identity of an operation, solve $a*e=a$ for $e$ and check that the same $e$ works for EVERY $a$ and on both sides ( $e*a=a$ too).; Groups and the Group Axioms › Identities and Inverses
Integral domains, and when $\mathbb Z_n$ is a field: An integral domain is a commutative ring with unity and no zero divisors.; Rings and Fields › Integral Domains and $\mathbb Z_p$
Irreducibility, and the root test in low degree: A non-constant $f\in F[x]$ is irreducible over $F$ if it is not a product of two polynomials of smaller degree in $F[x]$ .; Polynomials over a Field › Irreducible Polynomials
Isomorphism: A homomorphism $\varphi: G \to H$ is an isomorphism if it is also a bijection; then $G$ and $H$ are called isomorphic, written $G \cong H$ .; Homomorphisms and Isomorphisms › Isomorphisms
Isomorphism invariants: If $\varphi: G \to H$ is an isomorphism then $|G| = |H|$ , $G$ is abelian iff $H$ is, and $\operatorname{ord}(g) = \operatorname{ord}(\varphi(g))$ for every $g \in G$ — so the multiset of element orders, and every other property built purely from the group o…; Homomorphisms and Isomorphisms › Invariants That Tell Groups Apart
Kernel and image: For a homomorphism $\varphi: G \to H$ , the kernel is $\ker\varphi = \{g \in G : \varphi(g) = e_H\}$ and the image is $\operatorname{im}\varphi = \{\varphi(g) : g \in G\} \subseteq H$ .; Homomorphisms and Isomorphisms › Kernel and Image
Lagrange's theorem: If $G$ is a finite group and $H \le G$ , then $|H|$ divides $|G|$ .; Cosets and Lagrange's Theorem › Lagrange's Theorem
Left and right cosets: For a subgroup $H \le G$ and $g \in G$ , the left coset $gH = \{gh : h \in H\}$ and the right coset $Hg = \{hg : h \in H\}$ .; Cosets and Lagrange's Theorem › Cosets
Normal subgroup: A subgroup $N \le G$ is normal, written $N \trianglelefteq G$ , if $gNg^{-1} = N$ for every $g \in G$ , where $gNg^{-1} = \{gng^{-1} : n \in N\}$ .; Normal Subgroups and Quotient Groups › Normal Subgroups
Orbit and stabilizer: For $x\in X$ , the orbit of $x$ is $\operatorname{Orb}(x)=\{g\cdot x:g\in G\}\subseteq X$ , the set of everywhere $x$ can be moved to.; Group Actions › Orbits and Stabilizers
Order of an element: The order of $g \in G$ , written $|g|$ or $\operatorname{ord}(g)$ , is the smallest positive integer $k$ with $g^k = e$ (additively, the smallest $k$ with $kg = 0$ ); if no such $k$ exists, $g$ has infinite order.; A Gallery of Groups › The Order of an Element
Permutations and $S_n$: A permutation of $\{1, 2, \ldots, n\}$ is a bijection from that set to itself.; Permutation Groups › Permutations and Cycle Notation
Primitive elements: The nonzero elements of $\mathbb F_q$ form a CYCLIC group of order $q-1$ under multiplication.; Finite Fields › The Multiplicative Group Is Cyclic
Reading a group from its table: A Cayley table lists the elements of a finite group along the top and side, with entry $(i,j)=a_i*a_j$ placed in row $a_i$ , column $a_j$ .; Groups and the Group Axioms › Cayley Tables
Remainder and factor theorems: Remainder theorem: dividing $f$ by $x-a$ leaves remainder $f(a)$ .; Polynomials over a Field › Roots and the Factor Theorem
Second and Third Isomorphism Theorems: Let $N \trianglelefteq G$ .; Normal Subgroups and Quotient Groups › The Second and Third Isomorphism Theorems
Simple groups as building blocks, and $A_n$ for $n\ge5$: Recall (Section 3) that $G$ is simple if $|G|>1$ and its only normal subgroups are $\{e\}$ and $G$ .; The Sylow Theorems › Simple Groups and $A_5$
Simple groups, and the Sylow-counting strategy: A group $G$ with $|G|>1$ is simple if its only normal subgroups are $\{e\}$ and $G$ itself.; The Sylow Theorems › Proving a Group Is Not Simple
Subfields correspond to divisors: The subfields of $\mathbb F_{p^n}$ are exactly the fields $\mathbb F_{p^m}$ with $m\mid n$ , one for each divisor $m$ .; Finite Fields › Subfields of a Finite Field
Subgroups and the subgroup tests: A subset $H \subseteq G$ is a subgroup, written $H \le G$ , when $H$ is itself a group under the operation of $G$ : it contains the identity, is closed under the operation, and is closed under inverses.; Subgroups and Cyclic Groups › Subgroups and the Subgroup Tests
Subgroups correspond to intermediate fields: For a finite Galois extension $E/F$ with group $G$ , the maps $H\mapsto\{x\in E:\sigma(x)=x\ \forall\sigma\in H\}$ (the fixed field) and $K\mapsto\operatorname{Gal}(E/K)$ are inverse bijections between subgroups $H\le G$ and intermediate fields $F⊆ K…; Galois Theory › The Fundamental Theorem of Galois Theory
The characteristic of a ring: The characteristic of a ring with unity is the least $n\ge1$ with $\underbrace{1+1+\cdots+1}_{n}=0$ , or $0$ if no such $n$ exists.; Rings and Fields › Characteristic
The cycle trick: For a group $G$ acting on the positions $1,\ldots,n$ of a figure, and colorings with $k$ colors, $ $|\operatorname{Fix}(g)|=k^{c(g)},$ $where$ c(g) $is the number of cycles in$ g$'s disjoint cycle decomposition (Topic 4) of the positions, counting fixed poin…; Group Actions › Counting Colorings Up to Symmetry
The cyclic product criterion: $\mathbb Z_m \times \mathbb Z_n$ is cyclic — and then $\mathbb Z_m\times\mathbb Z_n \cong \mathbb Z_{mn}$ — if and only if $\gcd(m,n)=1$ .; Direct Products and Finite Abelian Groups › When Is a Product Cyclic?
The definition of a field: A field is a commutative ring with unity $1\ne0$ in which every nonzero element is a unit.; Rings and Fields › Fields
The definition of a group: $(G,*)$ is a group if $*$ is a binary operation on $G$ satisfying three further axioms: (associativity) $(a*b)*c=a*(b*c)$ for all $a,b,c\in G$ ; (identity) there exists $e\in G$ with $e*a=a*e=a$ for all $a\in G$ ; (inverses) for every $a\in G$ there exists $a…; Groups and the Group Axioms › The Group Axioms
The definition of a ring: A ring $(R,+,\cdot)$ is a set with two binary operations such that: $(R,+)$ is an abelian group (identity $0$ , inverses $-a$ ); multiplication is associative; and the distributive laws $a(b+c)=ab+ac$ and $(a+b)c=ac+bc$ hold.; Rings and Fields › Rings
The degree of an extension: An extension $E$ of $F$ is a vector space over $F$ ; its dimension is the degree $[E:F]$ .; Field Extensions › Degree and Bases
The dihedral group $D_n$: $D_n$ is the group of symmetries of a regular $n$ -gon — rotations and reflections, composed like functions — and it has order $2n$ : it is generated by a rotation $r$ of order $n$ and a reflection $s$ of order $2$ , subject to the relation $srs^{-1} = r^{-1}$ .; A Gallery of Groups › Symmetries of a Polygon: the Dihedral Groups
The direct product: For groups $G$ and $H$ , the direct product $G \times H$ is the set of pairs $\{(g,h) : g \in G,\ h \in H\}$ under the componentwise operation $(g_1,h_1)(g_2,h_2) = (g_1g_2,\ h_1h_2)$ .; Direct Products and Finite Abelian Groups › Direct Products
The Fundamental Theorem of Finite Abelian Groups: Every finite abelian group $G$ is isomorphic to a direct product of cyclic groups of prime-power order, $ $G \cong \mathbb Z_{p_1^{a_1}} \times \mathbb Z_{p_2^{a_2}} \times \cdots \times \mathbb Z_{p_k^{a_k}},$ $ and this list of prime powers (the elementary …; Direct Products and Finite Abelian Groups › The Fundamental Theorem of Finite Abelian Groups
The Galois group of an extension: The $F$ -automorphisms of $E$ form a group $\operatorname{Gal}(E/F)$ under composition.; Galois Theory › The Galois Group
The general linear group $GL(n,\mathbb R)$: The $n\times n$ invertible real matrices form a group under matrix multiplication, $GL(n,\mathbb R)$ : the identity is $I$ , and $A$ has an inverse exactly when $\det A \ne 0$ .; A Gallery of Groups › Matrix Groups
The group $(\mathbb Z_n, +)$: $\mathbb Z_n = \{0, 1, 2, \ldots, n-1\}$ under addition mod $n$ is a group of order $n$ : the identity is $0$ , and the inverse of $a$ is $n - a$ (taking $a = 0$ to its own inverse $0$ ).; A Gallery of Groups › Integers Mod n Under Addition
The group $U(n)$: $U(n) = \{a \in \{1, \ldots, n-1\} : \gcd(a, n) = 1\}$ under multiplication mod $n$ is a group of order $\varphi(n)$ (Euler's totient), the group of units mod $n$ .; A Gallery of Groups › Units Mod n: the Group U(n)
The normalizer formula for $n_p$: For a Sylow $p$ -subgroup $P \le G$ , the normalizer $N_G(P) = \{g \in G : gPg^{-1}=P\}$ is the subgroup fixing $P$ as a set under conjugation.; The Sylow Theorems › Counting Sylow Subgroups
The orbit-stabilizer theorem: For a finite group $G$ acting on $X$ and any $x\in X$ , $ $|G|=|\operatorname{Orb}(x)|\cdot|\operatorname{Stab}(x)|.$ $Equivalently,$ |Orb(x)|=[G:Stab(x)] $: the orbit size is the index of the stabilizer, so it always divides$ |G|$.; Group Actions › The Orbit-Stabilizer Theorem
The order of a permutation: The order of $\sigma \in S_n$ is the least positive integer $k$ with $\sigma^k = e$ the same notion of order as Topic 3, applied to the cyclic subgroup $\langle\sigma\rangle$ .; Permutation Groups › The Order of a Permutation
The order of an element, and the order formula: The order of $a \in G$ is the smallest positive integer $k$ with $a^k = e$ (additively, $ka = 0$ ); if no such $k$ exists, $a$ has infinite order.; Subgroups and Cyclic Groups › Orders in a Cyclic Group
The polynomial $x^{p^n}-x$: Over $\mathbb Z_p$ , $x^{p^n}-x$ is the product of ALL monic irreducible polynomials whose degree divides $n$ , each appearing once.; Finite Fields › Counting Irreducible Polynomials
The quotient group $G/N$: For $N \trianglelefteq G$ , the quotient group $G/N$ is the set of cosets of $N$ with multiplication $(aN)(bN) = (ab)N$ .; Normal Subgroups and Quotient Groups › Quotient Groups
The ring $F[x]$ and the division algorithm: For a field $F$ , $F[x]$ is the ring of polynomials $a_nx^n+\cdots+a_1x+a_0$ with coefficients in $F$ .; Polynomials over a Field › Polynomial Rings and Division
The size of a conjugacy class, and the class equation: $|\operatorname{cl}(a)| = [G:C_G(a)]$ : the size of a conjugacy class equals the index of its centralizer (proved in Section 5).; Conjugacy and the Class Equation › The Class Equation
The splitting field of a polynomial: A splitting field of $f\in F[x]$ is a smallest extension $E$ of $F$ in which $f$ factors completely into linear factors; it is generated over $F$ by the roots of $f$ .; Splitting Fields and Constructions › Splitting Fields
The three Sylow theorems: Let $|G| = p^am$ with $p$ prime and $p \nmid m$ , so $p^a$ is the largest power of $p$ dividing $|G|$ .; The Sylow Theorems › The Three Sylow Theorems
Three corollaries: For a finite group $G$ and $g \in G$ : (1) the order of $g$ divides $|G|$ , since $\langle g\rangle$ is a subgroup of order $\operatorname{ord}(g)$ ; (2) $g^{|G|} = e$ ; (3) if $|G|=p$ is prime, $G$ is cyclic, generated by any $g \ne e$ .; Cosets and Lagrange's Theorem › Consequences of Lagrange's Theorem
Two canonical forms: A finite abelian group has two standard descriptions.; Direct Products and Finite Abelian Groups › Elementary Divisors and Invariant Factors
Two irreducibility tests over $\mathbb Q$: Eisenstein: if $f=a_nx^n+\cdots+a_0\in\mathbb Z[x]$ and a prime $p$ divides $a_0,\ldots,a_{n-1}$ , does not divide $a_n$ , and $p^2\nmid a_0$ , then $f$ is irreducible over $\mathbb Q$ .; Polynomials over a Field › Eisenstein's Criterion and Reduction mod p
Two properties an operation may have: An operation $*$ is associative if $(a*b)*c=a*(b*c)$ for all $a,b,c$ , and commutative if $a*b=b*a$ for all $a,b$ .; Groups and the Group Axioms › Binary Operations
What ruler and compass can build: Starting from $0$ and $1$ , the numbers obtainable by ruler and compass are the constructible numbers: they form a field and are closed under square roots, and nothing else can be reached.; Splitting Fields and Constructions › Constructible Numbers