Why symmetry transformations must have inverses (i.e they must be bijective)?

Question

A symmetry transformation is a transformation which maps a geometric object to itself. And as far as i know, the definition of a group stems from their properties. (As the definition of the equivalence relations stems from the properties of equality: Since there is no other way of objectively defining concept of equality. Same holds true for the concept of symmetry)

So, combining any two symmetry transformation gives us another symmetry transformation, i.e. set of symmetry transformations is closed (which is very intuitive)

We have an identity element ($T_x = x$ kind of transformation, which also very intuitive). And we know that operation must be associative (since function composition is).

But then we say any symmetry transformation (or group element) must have an inverse (i.e must be a bijection). Why is this? This is not obvious at all. Can anyone please explain this?

Answer 1

According to Schwartzman's The Words of Mathematics:

symmetric (adjective), symmetry (noun): the first element is from Greek sun- "together with," from the Indo-European root ksun "with." The second element is from Greek metron "a measure." The Indo-European root is probably me- "to measure." Suppose two points are symmetric with respect to a line; if you measure the distance between one of the points and the line of symmetry, then "together with" that measurement you have simultaneously also measured the distance between the other point and the line of symmetry; the two distances are equal.

So the root of the word concerns how multiple things share the same measurement. This concept is itself symmetric under exchanging the things being measured. There is no preference for transforming $x\mapsto y$ over the reverse direction $y\mapsto x$; both should be equally possible.

Answer 2

One general concept of symmetry of a geometric object $X$ is that if $A,B \subset X$, then $A$ "looks like" $B$ if and only if there exists a symmetry $f : X \to X$ such that $f(A)=B$.

Examples of this abound, although the chief example is present in Euclidean geometry, where "looks alike" is taken to mean "is congruent to":

• If $A,B$ are triangles in the plane then $A$ is congruent to $B$ if and only if there exists a symmetry $f$ of the plane such that $f(A)=B$.

Whatever "looks alike" means in this kind of general situation, one should be able to use it like an equivalence relation:

• The reflexive law: Every $A \subset X$ looks like itself.
• The transitive law: For every $A,B,C \subset X$, if $A$ looks like $B$, and if $B$ looks like $C$, then $A$ looks like $C$.
• The symmetric law: For every $A,B \subset X$, if $A$ looks like $B$ then $B$ looks like $A$.

You can actually see the transitive and symmetric laws present in one of Euclid's "Common Notions", which I have seen translated into English as "things equal to the same thing are also equal to one another", and which is actually applied in Euclid where "equal" is interpreted as "congruent".

So how should a symmetry behave, in order to guarantee that the "looks like" relation is an equivalence relation?

These three simple laws of symmetry would be perfect:

• The identity is a symmetry
• The composition of two symmetries is a symmetry
• The inverse of a symmetry is a symmetry

In other words, symmetries form a group.

Euclid could almost have discovered group theory.