Introduction to Groups - Abstract Algebra | Lecture Notes

Introduction to Groups

1 mins read 184 words Topic 1 of 9

This topic covers the basic definitions and properties of groups in abstract algebra, including the concept of closure, associativity, identity element, inverses, and the group operation.

Introduction to Groups

Definition
- A group is a set G together with a binary operation * satisfying four axioms: closure, associativity, identity, and inverses.
The Four Axioms
- Closure: for all a, b in G, a*b is also in G.
- Associativity: (ab)c = a(bc) for all a, b, c in G.
- Identity: there exists e in G such that ae = ea = a for all a.
- Inverses: for every a in G, there exists a^-1 in G such that a*a^-1 = e.
Examples of Groups
- Integers under addition, non-zero rational numbers under multiplication, and the set of symmetries of a shape under composition.
Abelian Groups
- A group where the operation is also commutative (ab = ba for all a, b); named after mathematician Niels Henrik Abel.
Order of a Group
- The number of elements in a group (finite groups) or the fact that it is infinite (infinite groups).
Why Groups Matter
- Groups formalize the idea of symmetry and structure, with applications in cryptography, physics, and chemistry alongside pure mathematics.

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