**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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