Group¶
Informally, a group \(G\) is any set equipped with an operation \((\cdot)\). A group must satisfy the following four properties:
- Closure: the product of two elements in \(G\) are also in \(G\), i.e., for all \(a,b ∈ G\), \(a \cdot b\) is also in \(G\).
- Associativity: for all \(a,b, c ∈ G, a \cdot (b \cdot c) = (a \cdot b) \cdot c\).
- Identity: there an element denoted \(1_G ∈ G\) such that \(1_G \cdot g = g \cdot 1_G = g\) for all \(g ∈ G\)
- Invertibility: For each \(g ∈ G\), there is an element \(h\) in \(G\) such that \(g \cdot h = 1_G\). This element \(h\) is denoted \(g^{−1}\) . An example is the the set of invertible matrices, which forms a group under the matrix multiplication operation.
A group \(G\) is said to be cyclic if there is some group element \(g\) such that all group elements can be generated by repeatedly multiplying \(g\) with itself, i.e., if every element of \(G\) can be written as \(g^i\) for some positive integer \(i\). Such an element of \(g\) is called a generator for \(G\). Any cyclic group is abelian.
The cardinality \(|G|\) is called the order of \(G\). For any integer \(ℓ\), if \(z ≡ ℓ\) mod \(|G|,\) then \(g^ℓ = g^z\) (and \(g^{|G|} = 1_G\)).
A subgroup of a group \(G\) is a subset \(H\) of \(G\) that itself forms a group under the same binary operation as \(G\) itself.