Groups: Binary operations

Understanding Binary Operations in Groups

• A group is mathematical structure consisting of a set of elements combined with an operation.
• The operation combining two elements in the group is known as a binary operation.
• This binary operation must follow a set of strict criteria to create a group.
• These criteria include closure, associativity, an identity element, and the inverse element.

Characteristics of Binary Operations

• For closure, if two elements a and b belong to a group G, then the result of their binary operation, typically denoted as a*b or ab, must also be within G.
• Associativity means that for three elements in the group G, a, b, and c, the operation should satisfy the following: (ab)c = a(bc).
• A group must contain an identity element, often denoted by letter ‘e’ or the number ‘1’. The identity element is a special number such that when any element a in the group G is combined with this identity element, the original number remains unchanged, i.e., ae = ea = a.
• Every element ‘a’ within the group G should have an inverse element, often denoted by a^(-1) or -a within G. When an element and its inverse are combined using the binary operation, the result is the identity element, i.e., aa^(-1) = a^(-1)a = e.

Subgroups and Binary Operations

• A subgroup is a smaller group contained inside a larger group. It is also a set of elements combined with the same binary operation.
• A subgroup must satisfy the same group axioms: closure, associativity, identity, and inverse.

Applications of Binary Operations and Group Theory

• Group theory, including binary operations, has wide applications in different fields such as quantum physics, chemistry, and cryptography.
• In quantum physics, group theory is used to develop quantum mechanics.
• Chemists use group theory to understand the bonding and behavior of molecules.
• In cryptography, group theory is used to create complex codes and decipher encrypted messages.

Examples of Binary Operations in Groups

• Familiar operations like addition (+) and multiplication (x) on the set of integers are examples of binary operations forming a group.
• Matrix multiplication is a binary operation that can form a group when combined with the set of invertible matrices.
• Function composition is a binary operation that forms a group with the set of bijections (one-to-one correspondences) from a set to itself.
• Union and intersection of sets can be seen as binary operations and form a group when combined with the set of all subsets of a given set (forming the power set).