Even and odd functions

Recall that a function 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 $R \to R$ : in other words, each element of the group describes a function of type

$(R \to R) \to (R \to R)$

meaning that it takes as input a function $R \to R$ and returns as output another function $R \to R$ .

$C_{2}$ 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:

$f (x) = \frac{f (x) + f (- x)}{2}      even + \frac{f (x) - f (- x)}{2}      odd .$

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.

¹_{That 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.}