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Advanced Mathematics — Abstract Algebra

Groups, rings, fields, and their structural properties

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galois_theory_g 24 terms Jan 21, 2026
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Terms 24

1
Group
Set G with binary operation satisfying closure, associativity, identity, and inverses
2
Abelian Group
Group where operation is commutative: a*b = b*a for all elements
3
Order of Group
|G|; number of elements; order of element g is smallest n with g^n = identity
4
Subgroup
Subset H of G closed under group operation and inverses; inherits group structure
5
Lagrange's Theorem
Order of subgroup divides order of group: |H| divides |G|; corollary: element order divides |G|
6
Normal Subgroup
H ⊴ G: gHg⁻¹ = H for all g; invariant under conjugation; allows quotient group
7
Quotient Group (G/H)
Set of cosets {gH}; group structure if H normal; |G/H| = |G|/|H|
8
Homomorphism
Structure-preserving map between groups: φ(ab) = φ(a)φ(b)
9
Isomorphism
Bijective homomorphism; groups are isomorphic if structurally identical
10
Kernel
ker(φ) = {g: φ(g) = e}; always normal subgroup of domain
11
First Isomorphism Theorem
G/ker(φ) ≅ image(φ); quotient by kernel is isomorphic to image
12
Symmetric Group Sₙ
Group of all permutations of n elements; |Sₙ| = n!; non-abelian for n≥3
13
Cyclic Group
Generated by single element: G = ⟨g⟩ = {g^n: n∈ℤ}; isomorphic to ℤₙ or ℤ
14
Ring
Set with two operations (+ and ×); forms abelian group under +; × associative and distributive
15
Integral Domain
Commutative ring with no zero divisors: ab=0 ⟹ a=0 or b=0
16
Field
Commutative ring where every nonzero element has multiplicative inverse; e.g. ℚ, ℝ, ℂ, ℤ_p
17
Ideal
Subset I of ring R closed under addition and multiplication by ring elements; allows quotient ring
18
Principal Ideal Domain (PID)
Integral domain where every ideal is generated by one element; e.g. ℤ, polynomial ring over field
19
Unique Factorization Domain (UFD)
Every element factors uniquely into irreducibles (up to order and units); PIDs are UFDs
20
Galois Theory
Correspondence between field extensions and groups of symmetries; explains solvability of polynomials
21
Galois Group
Group of automorphisms of field extension fixing base field; Gal(E/F)
22
Fundamental Theorem of Galois Theory
Bijection between subfields and subgroups of Galois group; reverses inclusion
23
Solvability by Radicals
Polynomial solvable by radicals iff its Galois group is solvable; degree ≥5 generally not solvable
24
Simple Group
Group with no proper normal subgroups; building blocks; largest sporadic: Monster group