Date: Tue, 27 Dec 94 11:23:06 -0500 (EST)
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ Subject: Normal Subgroups of G

Recently, there was some discussion of whether the set C of twenty-four
rotations is a normal subgroup of the cube group G=<Q>. It isn't, but
I decided to write up some information about normal subgroups as it
relates to the cube. Most of the following is from Frey and Singmaster.
Any good stuff is theirs. Any crud that sneaks in is mine.

If H is any subgroup of G, a right coset of H in G is a set {hX}
for some fixed X in G and for all h in H. Similarly, a left coset
of H in G is a set {Xh}. Right cosets may be denoted as Hx, and
left cosets by xH.

In general, a right coset Hx is not equal to a left coset xH. But if
we have Hx=xH for all x in G, then H is a normal subgroup of G. An
alternative definition is H is normal if x'Hx=H for every x in G.
The definitions are equivalent, and Frey and Singmaster give as a
theorem Hx=xH for every x in G if and only if x'Hx=H for every x in G.

It should be noted that H normal does not imply that the elements
h of H commute with the elements x of G. That is, just because
Hx=xH we do not necessarily have hx=xh for every h in H (or even for
any h in H other than the identity). However, I think it is fair
to characterize a normal subgroup as commuting "globally" with G,
even if it does not commute "locally". On the other hand, if a
subgroup H does commute "locally" (i.e., if hx=xh for all h in H
and all x in G), then H is certainly normal.

Normal groups serve a function with respect to finite groups analogous
to the function served by prime numbers with respect to natural numbers.
First of all, any finite group always has at least two trivial
normal subgroups, namely the group itself and the group containing
only the identity. Second, a finite group containing normal subgroups
may be "factored" in a fashion analogous to prime numbers factoring
composite numbers. A finite group containing no normal subgroups
is called simple, analogous to numbers with no factors being called
prime.

The cube group G does not have very many normal subgroups, but it does
have a few. The first place to look for normal subgroups is to look
for subgroups with index 2. That is, look for subgroups that are
half as big a G. Such a subgroup is the subgroup A of even
permutations. ("A" stands for "Alternating", I think.)

It is easy to see that A is normal. If x is even, then Ax=xA=A.
if x is odd, then Ax=xA=Abar, where Abar is the set (not group!)
of odd permutations.

Similarly, any subgroup H with index 2 is normal. If the index of H
in G is 2, then H partitions G into two equal size sets H and Hbar.
If x is in H, then Hx=xH=H. If x is in Hbar, then Hx=xH=Hbar.

If we may digress briefly to the set M of 48 rotations and reflections,
then there are three subgroups of M with index 2. In Dan Hoey's
taxonomy, they are called C, A, and H. We may categorize the elements
of M as even or odd, and as rotations or reflections. There are 12
even rotations, 12 odd rotations, 12 even reflections, and 12 odd
reflections. If we take 12 even rotations and 12 odd rotations,
we have C. So C is a normal subgroup of M, even if it is not a normal
subgroup of G. If we take 12 even rotations and 12 even reflections,
we have A. This A (a subgroup of M) is not to be confused with the
A we have already talked about which is a subgroup of G. But I think
the name derives from the same source ("Alternating") in either case.
If we take 12 even rotations and 12 odd reflections, we have H.

Returning to G, the next two normal subgroups are Ac which leaves
the set of edges fixed, and Ae which leaves the set of corners
fixed. Ac is even on the corners, and Ae is even on the edges, in
order to conserve parity. Note that both Ac and Ae are normal
subgroups of A as well as of G.

I suppose that what is going on with Ac and Ae is obvious enough,
but I want to talk about it for a minute anyway. I most typically
think of an equation such as X=RLUD'R as meaning something to the
effect that "X" is a shorthand *name* for the collection of five
processes (in order) R, L, U, D', and R. But I still tend to think
of the processes as distinct. However, from the point of view
of group theory, X is a single operation which exists in its own
right just as do the quarter turns.

With a physical cube, you cannot perform an operation in Ac or Ae
without making a fairly long sequence of quarter turns. For
example, something so simple as performing FF on the corners while
leaving the edges fixed is non-trivial. But from the point of view
of group theory, we can easily find a single permutation
X[C,E] such that X[C]=FF[C] while X[E]=I[E]. Indeed, from the
point of view of group theory, you are never more than one move
from Start. That is, if you are at X, the one move which will always
solve the cube is X'. It is only if you are asked to decompose X'
into generators such as quarter-turns that the question of how far
from Start you are makes any sense.

If a subgroup H of G is normal, the left cosets form a group under the
operation (xH)(yH)=(xy)H. This group is called the factor group
of H in G or the quotient group of G by H, and is denoted as G/H.

Martin Schoenert recently clarified that while there may be more than
one way to define an operation on cosets such that they form a group,
the notation G/H is usually reserved for the case where the operation
is (xH)(yH)=(xy)H.

The factor group G/A contains two elements, and is isomorphic to any
group containing only two elements. We may write it as
<Abar>={A,Abar}, where A is the identity of the group.

The factor group G/Ac is isomorphic to the set of all permutations
on the edges (which we have written as G[E] in the recent past).
The factor group G/Ae is isomorphic to the set of all permutations
on the corners (which we have written as G[C] in the recent past).

Since Ac and Ae are normal subgroups of A, we may write A/Ac and A/Ae
which are isomorphic to Ae and Ac, respectively.

We can find normal subgroups of Ac and Ae. The set At of all
permutations in Ac which leave all corner locations fixed except for
twisting some of them is a normal subgroup of Ac. The set Af of
all permutations in Ae which leave all edge locations fixed except
for flipping some of them is a normal subgroup of Ae. (This twists
and flips have to follow the normal rules of conservation of twist
and flip, of course.)

This completes the list of normal subgroups. I will now give Frey
and Singmaster's proof that we are done, while interposing some
questions of my own for the cube theory experts out there. My
first question is that Frey and Singmaster do not state that At and
Af are normal subgroups of G. It seems obvious that they are.
However, is the formal argument that (for example) At is a normal
subgroup of Ac and Ac is a normal subgroup of G; hence, At is a
normal subgroup of G? How analogous is the factoring of groups
by normal subgroups to the factoring of composite numbers by
prime numbers?

Continuing with Frey and Singmaster, we may write Ac/At and
Ae/Af, where Ac/At is isomorphic to the group Asc which leaves
the corners sane and Ae/Af is isomorphic to the group Ase which
leaves the edges sane. "Sane" is a term used by Frey and Singmaster
in their proof of conservation of twist and flip. In general, it
is easy to see if a cubie is twisted or flipped when it is home,
but it is not so easy to see if it is twisted or flipped when it
is not home. Their proof (and the others I have seen) define a
frame of reference so that you can tell if a cube is twisted or
flipped when it is not home. A cubie which is not twisted or
flipped in this frame of reference is sane.

Asc and Ase are not normal subgroups of Ac and Ae, respectively.
(I tend to think that the reason they are not normal is related
to the fact that the frame of reference required to define sane
positions is not unique.) However, Asc and Ase are isomorphic
to well known groups.

The group Sn of all permutations of n objects is the n-element
symmetric group. The subgroup An of all even permutations of
n objects is the n-element alternating group (there is that word
"alternating" again!). Asc is isomorphic to A8 (there being
eight corner cubies) and Ase is isomorphic to A12 (there being
twelve edge cubies).

A famous result from Abel and Galois is that An does not have any
non-trivial normal subgroups for n >= 5. Hence, we have reduced
G to normal subgroups which have no more normal subgroups, and we
are done.

I guess my questions are as follows: 1) why must we restrict ourselves
to alternating groups? 2) For example, just as we found three
subgroups of M with index 2, might we not find other subgroups of
G with index 2 than the one we found? 3) Might we not find a
normal subgroup of G with some index other than 2, e.g., with index 3?

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Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax