Day 06Part 05- Discrete Mathematics for gate computer science – Special Properties of group.

Day 06Part 05- Discrete Mathematics for gate computer science – Special Properties of group.

​In Discrete Mathematics, particularly within the context of GATE Computer Science preparation, understanding the special properties of groups is essential. A group is a set combined with a binary operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. Beyond these, groups exhibit additional noteworthy properties:​

1. Uniqueness of Identity: Each group has a single, unique identity element.​

2. Uniqueness of Inverses: Every element in a group has a unique inverse.​

3. Cancellation Laws:

   • Left Cancellation: If $a\cdot b=a\cdot c$, then $b=c$.​

   • Right Cancellation: If $b\cdot a=c\cdot a$, then $b=c$.​

4. Order of an Element: The smallest positive integer $n$ such that an=ea^n = ean=e (where $e$ is the identity element) is called the order of the element $a$.​

5. Order of a Group: The total number of elements in the group. For finite groups, this is a positive integer.​

6. For a more in-depth understanding and visual explanation of these properties, you might find the following video lecture helpful:​