Groups

Groups are essential structures in the world of mathematics, with a special significance in abstract algebra.

A group $(G, \circ)$ is essentially a collection of elements that conforms to four distinct properties related to a binary operation (commonly symbolized by \(\circ\)). This operation combines any two elements from the collection to produce a third element also within the same collection.

So, in a more formal sense, a group $(G, \circ)$ is a combination of a non-empty set $G$ and a closed binary operation $ \circ : G \times G \rightarrow G $.

The four properties that characterize a group are:

• Closure
  For every pair of elements \(a\) and \(b\) in \(G\), the outcome of the operation \(a \circ b\) is also in \(G\). $$ \circ : G \times G \rightarrow G $$
• Associativity
  For any \(a\), \(b\), and \(c\) in \(G\), the following relationship holds true: $$ (a \circ b) \circ c = a \circ (b \circ c) $$

  It's worth mentioning that commutativity is not a necessary condition for a group. If the operation $ \circ $ in the group is commutative, meaning if \(a \circ b = b \circ a\) for all \(a, b \in G\), then the group is referred to as an abelian group (or a commutative group).

• Identity Element
  There exists an element \(e\) in \(G\), known as the identity element, such that for every element \(a\) in \(G\), combining \(a\) with \(e\) leaves \(a\) unchanged, that is $$ e \circ a = a \circ e = a $$
• Inverse Element
  For every element \(a\) in \(G\), there is an element \(a'\) in \(G\) such that $$ a \circ a' = a' \circ a = e $$ where \(e\) is the identity element.

Groups provide a framework for exploring symmetries and have applications across a wide range of fields in mathematics, physics, chemistry, computer science, and beyond.

For example, groups can describe the symmetry in geometric structures, symmetries of differential equations, symmetry operations in particle physics, and many other symmetry-related and conservation concepts.

An Example of a Group

A simple and very common example of a group is the set of integers \(\mathbb{Z}\) with the addition operation \(+\).

Let's verify the properties:

• Closure
  If you take any two integers, their sum is still an integer.

  For instance, \(3 + 5 = 8\), and \(8\) is an integer.

• Associativity
  Integer addition is associative.

  For example, \((2 + 3) + 4 = 2 + (3 + 4)\).

• Identity Element
  Zero \(0\) is the identity element for addition, as any integer added to \(0\) remains unchanged.

  For instance, \(5 + 0 = 5\).

• Inverse Element
  Every integer \(n\) has an inverse which is \(-n\), because \(n + (-n) = 0\).

  For example, the inverse of \(3\) is \(-3\), since \(3 + (-3) = 0\).

Additionally, since integer addition is commutative, this group is also an abelian group.