Group
Suppose \(G\) is a set, and \(*\) is a binary operator \(* : G \times G \rightarrow G\)
Then \((G, *)\) is a group if
- \(*\) is closed binary operator. So we have:
- Associativity:
- Identity: \(G\) has identity element \(e\).
- Inverse: Every element in \(G\) have an inverse.
Abelian (Commutative) Group
\((G, *)\) is a Abelian group if satisfies the Commutativity property:
\[\forall a, b \in G, \space \space a * b = b * a. \]Notation
We will denote the multiplicative and additive shorthand for iterated group operations by default. That is,
- for \(g \in (G, \times), \space g^2 = g * g, \space g^3 = g * g * g\), and so forth. \(g^{-1}\) for inverse. \(1\) instead of identity element \(e\).
- for \(g \in (G, +), \space 2g = g * g, \space 3g = g * g * g\), and so forth. \(-g\) for inverse. \(0\) instead of identity element \(e\).
Order of a group
- The order of a group is the number of elements in it, namely \(|g|\).
- If the group \(G\) has finite element we say \(G\) is finite group.
Order of an element in group
For \(g \in G\), the minimum integer \(n\) such that \(g^n = e\) called order of g.
Generator
- Any element \(g \in G\) that is able to generate \(G\) is called a generator.
- Of course, every group G has a trivial set of generators, when we just consider every element of the group to be in the generator set
Cyclic groups
Groups with single, not necessarily unique, generators are called cyclic groups.