Editing Abstract Algebra (section)

== <span style="color: #FFFFFF;">Analyzing</span> ==
{| class="wikitable"
|+ Algebraic Structures Compared
! Structure !! Operations !! Axioms !! Examples
|-
| Group || 1 binary op || Closure, assoc, identity, inverses || ℤ under +; permutations; rotations
|-
| Abelian group || 1 commutative op || Group + commutativity || ℤ, ℚ, ℝ under + ; ℤ/nℤ
|-
| Ring || + and × || Abelian group (+); monoid (×); distributive || ℤ, ℤ[x], Mn(ℝ), ℤ/nℤ
|-
| Field || + and × || Ring + every nonzero element invertible || ℚ, ℝ, ℂ, 𝔽''p, 𝔽''{p^n}
|-
| Vector space || + and scalar × || Abelian group (+); scalar mult axioms || ℝⁿ, Cⁿ, polynomial spaces
|}

'''Fundamental theorems''': Lagrange's Theorem (subgroup orders divide group order). First Isomorphism Theorem (G/ker(φ) ≅ Im(φ)). Sylow's Theorems (existence of prime-power subgroups). Fundamental Theorem of Finitely Generated Abelian Groups (every such group is a direct product of cyclic groups). Classification of Finite Simple Groups (completed 2004).
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