Lesson 14 — Advanced

Groups, Rings & Fields

Abstract algebra sabko darata hai, par andar se ye bahut simple hai — sirf checklist check karni hai. Is lesson mein seekhenge group ke 4 axioms, ring aur field ki definitions, aur examples se pehchanna kya group hai aur kya nahi. GATE mein 2 marks ka fixed topic hai.

Group kya hai — 4 axioms

Group ka matlab sirf "kuch operations hain" nahi — 4 conditions mil kar chalni chahiye. Ek bhi fail hui toh group nahi. Ye 4 conditions exam mein directly poochhi jaati hain.

Axioms checklist aur examples

4 AXIOMS (checklist):
  1. Closure       : a ∗ b bhi G mein hi ho
  2. Associativity : (a ∗ b) ∗ c = a ∗ (b ∗ c)
  3. Identity      : ek e jiske saath a ∗ e = e ∗ a = a
  4. Inverse       : har a ke liye a⁻¹, a ∗ a⁻¹ = e

Examples:
  (Z, +)        : GROUP ✓   identity 0, inverse of a = −a
  (Z, ×)        : group NAHI (2 ka koi integer inverse nahi)
  (Z_n, + mod n): GROUP ✓   identity 0, inverse of a = n − a
  (Q*, ×)       : GROUP ✓   identity 1, inverse of a = 1/a
  (N, +)        : NAHI      (1 ka inverse −1, natural nahi)
  (Z, −)        : NAHI      (5−3)−2 = 0 ≠ 5−(3−2) = 4
                  — associativity fail!

• Pehle closure check karo — sabse aasan fail point (5 − 7 = −2, Z mein hai, N mein nahi).
• Identity unique hoti hai — ek structure mein do identity nahi ho sakti.
• Har group abelian nahi hota — matrix groups, rotations non-commutative hote hain.
• Order of group: elements ki ginti. Finite group ka order fixed hota hai.

Ring kya hai

Ring do operations (+ aur ×) wala structure hai — jaise integers. Ismein + ek group banata hai, × sirf associative:

RING (R, +, ×) ki conditions:
  1. (R, +) abelian group hai  → + ke inverses chahiye
  2. (R, ×) semigroup hai     → closure + associativity
  3. Distributive             → a × (b + c) = a×b + a×c

Commutative ring : × commutative ho
Ring with unity  : multiplicative identity 1 ho
  (Z, +, ×) = commutative ring with unity ✓

SUBGROUP : group ka khud ka subset jo khud group ho
Lagrange : subgroup ka order, group ke order se
           DIVIDE hota hai
  (order = elements ki ginti; jaise order 6 ke group
   mein subgroup ka order 1, 2, 3 ya 6 hi ho sakta hai)

Lagrange theorem bahut powerful hai — group mein order kisi element ka bhi group ke order se divide karega (element ka order = usse identity tak pahunchne ke steps).

Field kya hai

Field sabse complete structure hai — multiplication mein bhi inverses chahiye (0 ke alawa):

FIELD = commutative ring with unity jahan har NON-ZERO
element ka MULTIPLICATIVE inverse bhi ho

(Q, +, ×)  ✓   2 ka inverse 1/2
(R, +, ×)  ✓   reals ka poora system
(C, +, ×)  ✓   complex numbers bhi field hain
Z (integers) FIELD NAHI — 2 ka 1/2 integer nahi

Z_p (p prime) = FIELD ✓  (mod p arithmetic)
  Z_5 : 2⁻¹ = 3   (2 × 3 = 6 = 1 mod 5) ✓

Z_6 FIELD NAHI — 2 × 3 = 0 (mod 6)
  (zero divisor! 2 ka inverse exist hi nahi sakta)

Yaad: prime modulus = field, composite = nahi

Exercise

Common mistakes

• Group mein commutativity zaroori samajhna: Commutativity sirf ABELIAN group ke liye hai. Group ke liye sirf 4 axioms: closure, associativity, identity, inverse. Rotations jaise groups commutative nahi hote.
• Identity aur inverse confuse karna: Identity ke saath operation = wahi element milta hai (a ∗ e = a). Inverse ke saath = identity milti hai (a ∗ a⁻¹ = e). Dono alag cheezein hain.
• Z ko field samajhna: Integers mein fractions wale inverses nahi hain — Z ek commutative ring hai, field nahi. Prime modulus wale Z_p fields hain.

5 baje + 8 ghante = 13 = 1 (mod 12) → ghadi pe 1 baje
(Z_12, +) = group  (identity 0, inverse of a = 12 − a)
(Z_12, ×) = group NAHI (6 ka inverse nahi, 6 × 2 = 0)
Sirf coprime wale: U(12) = {1, 5, 7, 11}
  5×5 = 25 = 1,  7×7 = 49 = 1,  11×11 = 121 = 1 ✓