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Group Theory

Group theory is a branch of mathematics that deals with the study of symmetry and abstract algebraic structures called groups. A group is a set of elements together with an operation that satisfies certain axioms. The study of groups has applications in various fields such as physics, chemistry, cryptography, and computer science.

Definition of a Group

A group is a set G together with a binary operation * that satisfies the following axioms:

  1. Closure: For all a, b in G, a * b is also in G.
  2. Associativity: For all a, b, c in G, (a * b) * c = a * (b * c).
  3. Identity: There exists an element e in G such that for all a in G, a * e = e * a = a.
  4. Inverse: For all a in G, there exists an element a^-1 in G such that a * a^-1 = a^-1 * a = e.

Examples of Groups

  1. The set of integers under addition is a group. The identity element is 0 and the inverse of an integer a is -a.
  2. The set of nonzero real numbers under multiplication is a group. The identity element is 1 and the inverse of a real number a is 1/a.
  3. The set of 2x2 invertible matrices under matrix multiplication is a group. The identity element is the 2x2 identity matrix and the inverse of a matrix A is its inverse matrix A^-1.

Subgroups

A subgroup H of a group G is a subset of G that is also a group under the same operation * as G. The subgroup H must satisfy the closure, associativity, identity, and inverse axioms.

Homomorphisms

A homomorphism is a function between two groups that preserves the operation. That is, if f: G -> H is a homomorphism, then f(a * b) = f(a) * f(b) for all a, b in G.

Isomorphisms

An isomorphism is a bijective homomorphism. That is, if f: G -> H is an isomorphism, then f is a one-to-one and onto function, and f(a * b) = f(a) * f(b) for all a, b in G. If there exists an isomorphism between two groups, then the groups are said to be isomorphic, denoted by G ≅ H.

Conclusion

Group theory is a fundamental topic in mathematics with broad applications in various fields. The notion of a group allows us to study symmetry in a rigorous and systematic way. The concepts of subgroups, homomorphisms, and isomorphisms further deepen our understanding of groups and their properties.

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