involution is normal

[PP]

Hello Group,

When a subgroup H of a group G exists out of {e,s} with s an involution,
then when I try with examples, I see that the subgroup H is normal in G.
But how could I proof this?

Regards

[Tonico]

What do you mean by "when a sbgp. H exists ""out"" of {e,s},..." ??
Do you mean that H = {e,s} is a sbgp. ? Then, obviously, s is an
involution.
Is your question why this is a normal sbgp.? If so then the answer is:
no, it usually is not a normal sbgp. ( counterexample: H = {(1),(12)}
in S_3 )
If you meant to ask something else then discard the above.

Tonio

[PP]

Tonico schreef:

Hello Tonio,

Yes, I meant to ask why this is a normal sbgp. It is now obvious wrong.
The case I was looking at is this:

In the icosahedron the involution that swaps opposite points is normal
in the automorfisme group of the icosahedron.

It works in all the cases, but I was wondering if there is an elegant
way to proof this. Now, as the number of cases is finite, I can just try
them all.

Regards

[Tonico]

> Regards-

I'm not sure I understand: the symmetry group of the icosahedron (and
this is what I understand by "automorphism group of the icosahedron",
as you wrote) is the alternating group A_5 , which is a simple group
and thus has no non-trivial normal subgroups...unless you meant
something else.
A reference here to a book where you read the above could be useful.

Tonio

[Jack Schmidt]

> In the icosahedron the involution that swaps opposite
> points is normal in the automorfisme group of the
> icosahedron.

You might have better luck proving that involution is
always central (directly, without worrying about normal
subgroups of order 2).

The matrix of the involution is -I where I is the identity
(assuming you center your figures at 0). Every matrix
commutes with -I, including the matrices of all the
isometries of the regular solids.

[Jack Schmidt]

> I'm not sure I understand: the symmetry group of the

> icosahedron is the alternating group A_5.

That is the rotational symmetry group. If you allow
reflections as well (orientation reversing) then the
symmetry group has order 120. It is a very boring
group A5 x C2, and so "icosahedral group" refers only
to the rotational part.

In terms of the original question, this just mean the
icosahedron part is mostly irrelevant. The involution
he is interested in is x -> -x which commutes with all
of the elements of the orthogonal group (the group of
all symmetries of all space that fix some specified
point). In particular, it is in the center of all
of the subgroups that contain it.

[PP]

Jack Schmidt schreef:

Thank you. This is a lot more elegant than my explication.