Elementary Group Theory | Mathematics Class 12

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Elementary Group Theory | Mathematics Class 12

Theory | Reference Notes | Elementary Group Theory | Subject: Mathematics Grade XII

1. Algebraic Structures: An algebraic structure of a set G under an operation on G and denoted by (G, ) satisfy the following properties.

  • Closed = if a, b ∊ G ⇒ a * b ∊ G
  • Commutative = if a * b = b * a, for each a, b ∊ G
  • Associative = if (a * b) * c = a * (b * c), for each a, b, c ∊ G
  • Identity = if a * e = e * a, for e ∊ G
  • Inverse = if a * b = e = b * a ⇒ b is inverse of a

2. Group: An algebraic structure (G, ), where G is a non-empty set with an operation ‘ ’ defined on it, is said to be group, if the operation satisfies the following axioms (called group axioms).

  • Closure Axiom → a * b ∊ G for all a, b ∊ G
  • Associative Axiom → (a * b) * c = a * (b * c) for all a, b, c ∊ G
  • Identity Axiom → a * e = e * a, for e, a ∊ G
  • Inverse Axiom → a * b = e = b * a, where a and b ∊ G, b is the inverse of a

3. Congruent Modulo m: If a positive integer a is divided by a positive integer m giving the positive integers k and b (<m) as the quotient and the remainder respectively then we call ‘a congruent b modulo m’ and I written as a = b (mod. m).

Cayley’s table

+3 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1

 

2 + 1 = 0 (mod. 3)

2 + 2 = 3 + 1 = 1 (mod. 3)

 

X3 0 1 2
0 0 0 0
1 0 1 2
2 0 2 1

 

2 X 1 = 2 (mod.3)

2 X 2 = 4 – 1 (mod. 3)

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