Dan Hoey writes in his e-mail message of 1994/11/08
What is the difference between a direct product and a semidirect product?
Allow to answer this in greater detail, and describe all the important
products of groups. It is one of the marvels of Rubik's cube that all
these products arise in a very natural fashion when one investigates it.
Direct Product
--------------
We say that a group D is the *direct product* of its two subgroups
M and N, if M and N together generate the whole group D, M and N
have trivial intersection, and M and N are both normal subgroups.
This implies that D is isomorphic to the group of pairs (m,n),
where the multiplication is defined componentwise,
i.e., (m_1,n_1) * (m_2,n_2) = (m_1 * n_2, m_1 * n_2).
Here is a little picture to describe the situation.
D
/ \
/ \
M \
\ N
\ /
\ /
1
M and N are the factors, D is their direct product.
Note that D/N ~ M and D/M ~ N (X ~ Y means that X and Y are isomorphic).
So M and N appear both as subgroups and quotients of the direct product.
Direct products are very simple. They are fully described by their
to factors M and N. Also many properties follow easily from the
corresponding properties of the factors, e.g.., |M*N| = |M| * |M|.
For an example lets take a group G+ that is a little bit larger than the
group G of Rubik's cube, namely we also allow to exchange two edges
*without* exchanging two corners simultaneously (don't ask me how this
could be realized physically). This group is the direct product of two
subgroups GC and GE. GC is the stabilizer of the edges, i.e., the
subgroup of those elements that do not permute the edges at all, and thus
operates only on the corners. GE is the stabilizer of the corners, i.e.,
the subgroup of those elements that do not permute the corners, and thus
operates only on the edges. G+ is isomorphic to the group of pairs (c,e)
with c in GC and e in GE. That means, each element of G+ can be
described by describing how it operates on the corners and on the edges
(this is trivial). And for any c in GC and any e in GE, there is an
element in G+ that operates like c on the corners and like e on the edges.
The latter is not true in G, which is a subgroup of index 2 in G+.
Semidirect Product
------------------
We say that a group S is the *semidirect product* of its two subgroups
H and N, if H and N together generate the whole group S, H and N have
trivial intersection, and N is a normal subgroup (but H need not be
normal). This implies that S is isomorphic to the group of pairs (h,n),
where the multiplication is defined as follows
(h_1,n_1) * (h_2,n_2) = (h_1 * h_2, h_2^-1 * n_1 * h_2 * n_2).
Note that h^-1 * n * h is usually written as n ^ h.
A picture for this situation would be identical to the picture for the
direct product, since the only real difference to the direct product is
that H need not be normal.
Again both H and N appear as subgroups of S, but only H appears as
factor of S, namely S/N ~ H. Note that S/H is *not* well defined,
since H is not a normal subgroup of S.
Semidirect products are almost as simple as direct products. They
are described by their factors H and N, and the operation of H on N,
by specifying what n^h is for every h in H and n in N
(note that n^h is again in N, because N is a normal subgroup).
And again many properties of S can be easily calculated from the
corresponding properties of H and N.
For an example let S be the group generated by the six face turns and
rotations of the entire cube (or equivalently the six face turns and the
three middle slice turns). Let G be the subgroup of S generated by the
six face turns. Let C be the subgroup of S generated by the rotations of
the entire cube, which has size 24. Obviously C and G together generate
S. C and G have trivial intersection, since every element of G fixes the
orientation of entire cube, but only the trivial element in C fixes the
entire cube. Finally G is normal, since for each generator (face turn) g
of G and each h in H the element g^h is again in G (in fact it is again a
generator or an inverse of a generator). Thus S is a semidirect product
of C and G.
Subdirect Product
-----------------
Let M and N be two groups. Let D be the direct product of M and N.
Let f: M -> H and g: N -> H be two homomorphisms onto a group H.
Then the subgroup S = {(m,n) in D | f(m) = g(n)} of D is called a
*subdirect product* of M and N.
Again a little picture to describe the situation.
D
/ | \
/ | \
/ | \
/ S \
/ | \
/ | \
M | \
\ | N
\ S- /
\ / \ /
\ / \ /
M- \ /
\ N-
\ /
\ /
1
M and N are the two factors, D is their direct product.
S is the subdirect product for the equation f(u) = g(v).
M- is the kern of f, and N- is the kern of g.
Thus M/M- = M/kern(f) ~ image(f) = H = image(g) ~ N/kern(g) = N/N-.
S- is the direct product of M- and N- and is a subgroup of S
(namely the subgroup such that f(u) = g(v) = 1).
It is easy to see that S/N- ~ M and S/M- ~ N.
So M and N appear as quotients of S.
But note that M and N *do not* appear as subgroups of S!
Also note that S/N- ~ M and S/M- ~ N implies that S/S- ~ H.
Thus M and N have a common quotient H, and in the subdirect product
we have ``glued'' these two quotients together.
For an example lets again look at the direct product G+ of GC and GE.
I have already mentioned that G is a subgroup of index 2 in G+.
It is in fact a subdirect of GC and GE. Namely H = {-1,1},
f(c) is the parity (sign) of the permutation of the corners by c, and
g(e) is the parity (sign) of the permutation of the edges by e.
In other words, G is the subdirect product of GC and GE with
the condition that whenever we exchange two corners we also
exchange two edges.
This is in fact no coincidence. Whenever one has a permutation group
that has more than one orbit, it is a subdirect product of the operations
on the individual orbits.
Wreath Product
--------------
Let M be an abitrary group. Let H be a permutation group operating on
[1..n]. For h in H and i in [1..n], we write the image of i under h as
i^h. The *wreath product* W = M wr H, is the semidirect product of the
normal subgroup N = M^n (i.e., the direct product of n copies of M),
with the subgroup H, where H operates by permuting the components of N,
(i.e., (m_1,m_2,...,m_n)^h := (m_{1^h},m_{2^h},...,m_{n^h})).
For an example take the following permutation group W:
< ( 1, 2, 3), ( 4, 5, 6), ( 7, 8, 9), (10,11,12),
(13,14,15), (16,17,18), (19,20,21), (22,23,24),
( 1, 4)( 2, 5)( 3, 6),
( 4, 7,10,13,16,19,22)( 5, 8,11,14,17,20,23)( 6, 9,12,15,18,21,24) >.
The first 8 generators generate a direct product N of 8 cyclic groups of
size 3, which is a normal subgroup in G. The last 2 generators generate
a symmetric group of degree 8, operating on the 8 factors of the direct
factors of N. Thus W = C_3 wr S_8.
W operates on the set { {1,2,3}, {4,5,6}, ..., {22,23,24} }, called a
blocksystem for W. To describe what an element w in W does, we first say
how it operates on each block, and then how it permutes the blocks.
On the other hand, if we have a permutation group that has a blocksystem,
then this permutation group is a subgroup of the wreath product.
The group GC, operating only on the corners of the cube, is a subgroup of
index 3 in the above group W. For each element of GC, you first say how
it changes the orientation of the 8 corners (the C_3^8) and then how it
permutes the 8 corners (the S_8). The index 3 comes from the fact that
we cannot change the orientation of a single corner in GC; if we turn
one corner clockwise, we must turn another corner counterclockwise.
Have a nice day.
Martin.