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.


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