symmetries and symmetry groups

The most common way to introduce groups is to discuss the symmetries
of plane figures, i.e. euclidean transformations which map the figure to
itself. From the kleinian view, there is no reason why we should restrict
attention to euclidean transformations.

Suppose that G is a group of transformations of a set S, so that G gives
a geometry on S. In this geometry, a figure F is simply a subset of S.
We may consider elements of G which map F to itself.

Suppose that G is a group of transformations of a set S, and that F is a
subset of S. An element g of G is a G-symmetry of F if g(F) = F. The set
of G-symmetries of F is denoted by S(F,G).

This set contains much geometrical information about F, as we shall see.

Theorem SG1
If G is a group of transformations of a set S, and F is a subset of S, then
S(F,G) is a subgroup of G.

proof of theorem SG1

The subgroup S(F,G) is called the G-symmetry group of F. If H is any
subgroup of S, it also defines a geometry on S, and hence we have an
H-symmetry group S(F,H). Of course, the symmetry groups are related.

Theorem K3
If G is a group of transformations of a set S, H a subgroup of G, and F is
a subset of S, then S(F,H) is a subgroup of S(F,G).

Since S(F,H) is a group, by Theorem SG1, it is enough to show that it is
a subset of S(F,G).
Suppose that h is in S(F,H). Then h(F) = F. Since H is a subgroup of G,
we also have hεG. Thus hεS(F,G), as required.

At the moment, we can offer ony a rather contrived example to show that
the subgroups may be different, i.e. that S(F,H) may be a proper subgroup.

Example 1
Let F be the ray {(x,0) : x ≥ 0} in R2. Then it is easy to see that S(F,E(2))
contains only e, the identity, and r, reflection in the real axis. On the other
hand, S(F,S(2)) also contains the scalings sk(x) = kx, k > 0. Thus, S(F,E(2))
is a proper subgroup of S(F,S(2)).

Recall that two subsets F, F' of S are G-congruent if there is an element g
of G with g(F) = F'. Since the definitions of G-congruence and G-symmetry
are somewhat similar, it is not surprising that the concepts are related.

Theorem SG2
If F and F' are G-congruent subsets of S, then S(F,G) and S(F',G) are
conjugate subgroups of G. Indeed, if gεG is such that F' = g(F), then
S(F',G) = gS(F,G)g-1.

proof of theorem SG2

In particular, we observe that the G-symmetry groups of G-congruent subsets
will be isomorphic groups. Unfortunately, the converse is false. We can have
non-congruent figures with identical (and hence isomorphic) symmetry groups.

Example 2
In the picture on the right, ABCD is a rectangle, and P,Q,P,S are the mid-points
of AB,BC,CD,DA respectively. Then the quadrilaterals ABCD and PQRS each have
E(2)-symmetry group K = {e,h,v,r}, where e is the identity, h and v are the
reflections in the lines SQ and PR, and r is the half-turn about X.

On the other hand the Theorem does allow us to conclude that figures with
non-isomorphic G-symmetry groups must be incongruent in the G-geometry.

symmetry groups for affine conics

symmetry groups for projective conics

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