Group theory: Examples

Even and odd functions

Recall that a function \(f : \mathbb{R} \to \mathbb{R}\) is even if \(f(-x) = f(x)\), and odd if \(f(-x) = - f(x)\). A typical example of an even function is \(f(x) = x^2\) or \(f(x) = \cos x\), while a typical example of an odd function is \(f(x) = x^3\) or \(f(x) = \sin x\).

We can think about evenness and oddness in terms of group theory as follows. There is a group called the cyclic group \(C_2\) of order \(2\) acting on the set of functions \(\mathbb{R} \to \mathbb{R}\): in other words, each element of the group describes a function of type

$$ (\mathbb{R} \to \mathbb{R}) \to (\mathbb{R} \to \mathbb{R}) $$

meaning that it takes as input a function \(\mathbb{R} \to \mathbb{R}\) and returns as output another function \(\mathbb{R} \to \mathbb{R}\).

\(C_2\) has two elements which we’ll call \(1\) and \(-1\). \(1\) is the identity element: it acts on functions by sending a function \(f(x)\) to the same function \(f(x)\) again.\(-1\) sends a function \(f(x)\) to the function \(f(-x)\), which visually corresponds to reflecting the graph of \(f(x)\) through the y-axis. The group multiplication is what the names of the group elements suggests, and in particular \((-1) \times (-1) = 1\), which corresponds to the fact that \(f(-(-x)) = f(x)\).

Any time a group \(G\) acts on a set \(X\), it’s interesting to ask what elements are invariant under that group action. Here the invariants of functions under the action of \(C_2\) above are the even functions, and they form a subspace of the vector space of all functions.

It turns out that every function is uniquely the sum of an even and an odd function, as follows:

$$f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even}} + \underbrace{\frac{f(x) - f(-x)}{2}}_{\text{odd}}.$$

This is a special case of various more general facts in representation theory, and in particular can be thought of as the simplest case of the discrete Fourier transform, which in turn is a toy model of the theory of Fourier series and the Fourier transform.

It’s also interesting to observe that the cyclic group \(C_2\) shows up in lots of other places in mathematics as well. For example, it is also the group describing how even and odd numbers add1 (where even corresponds to \(1\) and odd corresponds to \(-1\)); this is the simplest case of modular arithmetic.

1That is: an even plus an even make an even, an odd plus an odd make an even, and an even plus an odd make an odd.