Re: isometries, and symmetry groups

On Mar 30, 10:46 pm, Narcoleptic Insomniac
<i_have_narcoleptic_insom...@xxxxxxxxx> wrote:
On Mar 30, 2008 7:43 PM CT, crossedproduct wrote:

If M is a (nonempty) set, we can consider the group
of all bijections M -> M.

Suppose X is a subset of M; does the set of all
bijections which leave X invariant, necessarily form
a group ?
(A bijection f : M -> M which leaves X invariant is
such that f(X) = X).

I'm pretty certain that it does.

I believe that a few more conditions are required to
gaurantee a group structure.

Given a set M and a subset X of M, the set
of all bijections f: M --> M such that f(X) = X
is indeed a subgroup of the group S(X) of all
bijections X --> X: it is closed under composition
and inversion, and it is not empty.

[... snip ...]

How does this differ from the symmetry group of X,
which consists of all *isometries* of M (when M is a
metric space) leaving X fixed ?

They are very different: just consider pretty much
any example.

It would seem that the difference lies in the fact that
Homeo(X) is a group regardless of the topological space X
being metric, whereas Isom(M) requires M be a metric
space -- it would appear that Homeo(X) is a little more
general than Isom(M).

Of course, that begs the question: Is Homeo(M) =~ Isom(M)
for every metric space M? Intuitively, I would like to
say yes, however, this matter is a little beyond me at
the moment.

Those two groups are not isomorphic in general.

Take for example a solid triangle T in the plane
whose all three sides are of different length:
then the only isometry T --> T is the identity,
but the group of auto-homeomorphisms T --> T is
huge.

In other words, why are isometries important to
studying (or defining, for that matter) groups of
transformations of plane figures, and not just those
bijections which leave them invariant?

Isometries are important when you are studying the
metric properties of a plane figure. If you are studying
other types of properties, then other groups become
interesting, in general.

By the way, you say `not just those bijections which leave
them invariant' but in general there will be many many many
more bijections leaving it invariant than isometries
of the figure...

-- m
.

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