Alternating groups are simple
Statement
For , the Alternating group (?)
, i.e., the group of all even permutations on
letters, is a simple non-Abelian group.
Related facts
- Finitary alternating groups are simple: The finitary alternating group on an infinite set is simple.
Facts used
- A5 is simple
- Normality satisfies transfer condition: The intersection of a normal subgroup of the whole group, with any subgroup, is normal in the subgroup.
- Even permutation IAPS is padding-contranormal:
is a contranormal subgroup inside
. More specifically, the even permutations fixing any particular letter form a contranormal subgroup of the group of all even permutations.
Proof
The proof is by induction on .
Base case
The case is dealt with separately, by a direct argument. For full proof, refer: A5 is simple
Induction step
We prove that for , if
is simple, then
is simple.
Given: is the group of even permutations on
.
is a normal subgroup of
.
To prove: or
is trivial.
Proof: Let denote the subgroup of
that stabilizes the letter
. Then, each
consists of the even permutations on
letters (the letters excluding
) and is hence isomorphic to
. Thus, each
is simple.
Now, since normality satisfies transfer condition, is normal in
for every
. By simplicity of
, either
contains
, or
intersects
trivially.
Suppose there exists for which
contains
. Then, by fact (3) stated above,
is contranormal inside
, i.e., its normal closure is
. Since
is normal, this forces
, and we are done.
Otherwise, is trivial for every
. Thus, no nontrivial element of
fixes any letter. Let's use this to show that
can have no nontrivial elements.
Suppose is nontrivial. Then, first observe that in the cycle decomposition of
, every element must be in a cycle of the same length
(otherwise, some power of
would fix a letter). Thus,
has
cycles each of size
, where
.
Now, if , then
. Choose an
and a double transposition
such that
fixes both
and
, but such that
does not commute with
(this is possible because
). Then,
is a nontrivial element of
fixing both
and
, giving the required contradiction.