: Abstract Algebra Formula Sheet

Groups - Definition

Group Axioms

A group $(G, \cdot)$ is a set with operation $\cdot$ satisfying:

Closure: $\forall a,b \in G: ab \in G$

Assoc: $(ab)c = a(bc)$ for all $a,b,c$

Identity: $\exists e \in G$ with $ae = ea = a$

Inverse: $\forall a \in G, \exists a^{-1}$ with $aa^{-1} = e$

Abelian: If also commutative ($ab=ba$), called abelian/commutative.

Subgroups

Definition & Criteria

$H \subseteq G$ is a subgroup ($H \le G$) if:

$H \ne \emptyset$

Closed under $G$'s operation

Contains inverses (or $H$ finite and closed)

One-Step Test

$H \le G \iff H \ne \emptyset$ and $ab^{-1} \in H$ for all $a,b \in H$

Generated Subgroups

$\langle S \rangle$ = smallest subgroup containing $S$

For single element: $\langle a \rangle = \{a^n : n \in \mathbb{Z}\}$

Trivial: $\{e\}$ and $G$ are always subgroups.

Cyclic Groups

Definition

$G$ is cyclic if $G = \langle g \rangle$ for some $g \in G$

Order

Order of $g$: $\text{ord}(g) = \min\{n > 0 : g^n = e\}$

If no such $n$, then $g$ has infinite order

Properties

Cyclic group of order $n$: $C_n \cong \mathbb{Z}_n$

Every subgroup of cyclic group is cyclic

Number of generators of $C_n$: $\phi(n)$ (Euler's totient)

$$\langle g \rangle \cong \mathbb{Z}/\text{ord}(g)\mathbb{Z}$$

Permutation Groups

Symmetric Group $S_n$

All bijections of $\{1,\ldots,n\}$, order $n!$

Cycle Notation

$(1\ 2\ 3)$ means $1 \to 2 \to 3 \to 1$

$(i\ j)$ is a transposition (swaps two elements)

Disjoint cycles commute

Key Facts

Every perm is product of transpositions

Sign: $\text{sgn}(\sigma) = (-1)^{\text{# transpositions}}$

$A_n$ = alternating group (even permutations), $|A_n| = n!/2$

$$\text{ord}((i_1 \cdots i_k)) = k$$

Cosets & Lagrange's Theorem

Cosets

Left coset: $aH = \{ah : h \in H\}$
Right coset: $Ha = \{ha : h \in H\}$

All cosets have same cardinality as $H$

Cosets partition $G$

Lagrange's Theorem

If $G$ finite and $H \le G$: $$|H| \text{ divides } |G|$$

Index: $[G:H] = |G|/|H|$ = number of cosets

Consequences

Order of element divides $|G|$

$a^{|G|} = e$ for all $a \in G$

Groups of prime order are cyclic

Normal Subgroups

Definition

$N \triangleleft G$ if $gNg^{-1} = N$ for all $g \in G$
(or equivalently: $gN = Ng$ for all $g$)

Equivalent Conditions

Left cosets = right cosets

$N = Ker(\phi)$ for some homomorphism

Quotient Groups

If $N \triangleleft G$: $$G/N = \{gN : g \in G\}$$

Operation: $(aN)(bN) = (ab)N$

Order: $|G/N| = [G:N] = |G|/|N|$

Abelian subgroup: Any subgroup of abelian group is normal.

Homomorphisms

Definition

$\phi: G \to H$ is a homomorphism if: $$\phi(ab) = \phi(a)\phi(b)$$

Kernel & Image

$\ker(\phi) = \{g \in G : \phi(g) = e_H\}$
$\text{im}(\phi) = \{\phi(g) : g \in G\}$

Basic Properties

$\phi(e_G) = e_H$

$\phi(g^{-1}) = \phi(g)^{-1}$

$\ker(\phi) \triangleleft G$

$\phi$ injective $\iff$ $\ker(\phi) = \{e\}$

Isomorphism Theorems

1st Isomorphism Theorem

$$G/\ker(\phi) \cong \text{im}(\phi)$$

2nd Isomorphism Theorem

If $H \le G, N \triangleleft G$: $$HN/N \cong H/(H \cap N)$$

3rd Isomorphism Theorem

If $N \triangleleft G$ and $N \le M \triangleleft G$: $$(G/N)/(M/N) \cong G/M$$

Correspondence Thm: Subgroups of $G$ containing $N$ correspond to subgroups of $G/N$.

Group Actions

Definition

Action: $G \times X \to X$, $(g,x) \mapsto g \cdot x$ satisfying: $$e \cdot x = x, \quad (gh) \cdot x = g \cdot (h \cdot x)$$

Orbits & Stabilizers

Orbit of $x$: $\text{Orb}(x) = \{g \cdot x : g \in G\}$
Stabilizer: $\text{Stab}(x) = \{g \in G : g \cdot x = x\}$

Orbit-Stabilizer Thm

$$|\text{Orb}(x)| \cdot |\text{Stab}(x)| = |G|$$

Number of orbits $= \frac{1}{|G|}\sum_{g \in G}|X^g|$

Examples

Conjugation: $g \cdot a = gag^{-1}$

Left mult: $g \cdot h = gh$

Sylow Theorems

Sylow $p$-Subgroup

Let $|G| = p^a m$ where $p \nmid m$. A Sylow $p$-subgroup has order $p^a$.

Sylow's Theorems

1st: Sylow $p$-subgroups exist
2nd: All Sylow $p$-subgroups are conjugate
3rd: Number of Sylow $p$-subgroups divides $m$ and $\equiv 1 \pmod{p}$

Applications

If $n_p = 1$, the Sylow $p$-subgroup is normal

Groups of order $pq$ (distinct primes): often have structure

Useful: Sylow theorems help determine group structure without finding all subgroups.

Rings - Definition

Ring Axioms

$(R, +, \cdot)$ is a ring if: $(R, +)$ is abelian group, $(R, \cdot)$ is semigroup, distributive laws hold

Key Types

Commutative: $ab = ba$ for all $a,b$

Unital: Has multiplicative identity $1 \ne 0$

Integral Domain: Comm, unital, no zero divisors

Field: Comm ring where every nonzero elt invertible

Examples

$\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}[i]$

$\mathbb{Z}_n, \mathbb{F}_p[x]$ (polynomials)

Ideals

Definition

$I \subseteq R$ is an ideal if: $(I, +)$ is subgroup and $ra, ar \in I$ for all $r \in R, a \in I$

Types of Ideals

Left ideal: $ra \in I$ for all $r, a$

Right ideal: $ar \in I$ for all $a, r$

Two-sided: Both left and right ideal

Generated Ideals

$(a) = \{ra : r \in R\}$ (principal ideal in commutative ring with unity)

$(a_1, \ldots, a_n)$ = ideal generated by $\{a_1, \ldots, a_n\}$

Quotient Rings & Homomorphisms

Quotient Ring

If $I \triangleleft R$ (two-sided ideal): $$R/I = \{r + I : r \in R\}$$ with $(r+I) + (s+I) = (r+s)+I$ and $(r+I)(s+I) = rs+I$

Ring Homomorphism

$\phi: R \to S$ preserves $+$ and $\cdot$: $$\phi(a+b) = \phi(a)+\phi(b), \quad \phi(ab) = \phi(a)\phi(b)$$

1st Isomorphism Thm for Rings

$$R/\ker(\phi) \cong \text{im}(\phi)$$

$\ker(\phi) = \{r \in R : \phi(r) = 0\}$ is an ideal

Integral Domains & Fields

Integral Domain

Commutative unital ring with no zero divisors: $$ab = 0 \implies a = 0 \text{ or } b = 0$$

Fields

Commutative ring where every nonzero element has multiplicative inverse

$\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_p$ (p prime)

Key Facts

Every field is an integral domain

Finite integral domains are fields

$\mathbb{Z}_n$ is field $\iff$ $n$ is prime

Characteristic of $R$: smallest $n > 0$ with $n \cdot 1 = 0$

Polynomial Rings

Definition

$R[x]$ = polynomials in $x$ over $R$: $$f(x) = a_n x^n + \cdots + a_1 x + a_0, \quad a_i \in R$$

Division Algorithm

If $F$ is field and $f, g \in F[x]$ with $g \ne 0$:

$$f = qg + r$$ where $q, r \in F[x]$ and $\deg(r) < \deg(g)$

Irreducibility

$p(x) \in F[x]$ irreducible if not product of lower degree polys

Root $\implies$ linear factor: if $p(a) = 0$, then $(x-a) | p(x)$

Over field: $F[x]$ is Euclidean domain, hence PID

UFD • PID • Euclidean

Unique Factorization Domain

Every nonzero non-unit factors uniquely (up to order & units) into irreducibles

Example: $\mathbb{Z}, F[x]$ for field $F$

Principal Ideal Domain

Every ideal is principal: $I = (a)$ for some $a$

Every PID is UFD

Example: $\mathbb{Z}, F[x]$

Euclidean Domain

Has Euclidean algorithm: exists $N: R \setminus \{0\} \to \mathbb{Z}_{\ge 0}$ with quotient-remainder

Every ED is PID (hence UFD)

Note: UFD $\not\implies$ PID (e.g., $\mathbb{Z}[x]$)

Field Extensions

Definition

$K/F$ is field extension if $F \subseteq K$ are fields

Algebraic vs Transcendental

Algebraic $\alpha$: $\exists p(x) \in F[x], p(\alpha) = 0$

Transcendental $\alpha$: $\not\exists$ poly with $p(\alpha) = 0$

Degree

$$[K:F] = \dim_F(K)$$ (as vector space over $F$)

Tower law: $[L:F] = [L:K][K:F]$

Minimal Polynomial

Monic irreducible poly in $F[x]$ of smallest degree with root $\alpha$

Galois Theory

Galois Group

$$\text{Gal}(K/F) = \{\sigma: K \to K \text{ automorphism} : \sigma|_F = \text{id}\}$$

Galois Extension

$K/F$ Galois if $|\text{Gal}(K/F)| = [K:F]$

Equiv: $K$ splitting field of separable poly over $F$

Fundamental Theorem

For Galois $K/F$: subgroups of $\text{Gal}(K/F)$ $\leftrightarrow$ intermediate fields $F \subseteq L \subseteq K$

Correspondence: $H \leftrightarrow K^H$ (fixed field)

Key: Normal subgroups correspond to Galois subextensions.