4 1. Sylow Theory
1.1. Theorem. Let H C G be a subgroup, and let ft be the set of right cosets
of H in G. Then
is isomorphic to a subgroup ofSym(ft). In
particular, if the index \G : H\ n, then G/coveciH) is isomorphic to a
subgroup of Sn, the symmetric group on n symbols.
Proof. The action of G on the set ft by right multiplication defines a
homomorphism 9 (the permutation representation) from G into Sym(O).
Since ker(#) = corec(i^), it follows by the homomorphism theorem that
G/covec{H) = 0(G), which is a subgroup of Sym(G). The last statement
follows since if \G : H| = n, then (by definition of the index) |fi| = n, and
thus Sym(fi) ^ Sn.
1.2. Corollary. Let G be a group, and suppose that H C G is a subgroup
with \G : H\ n. Then H contains a normal subgroup N of G such that
\G : N\ divides n\.
Proof. Take T V = corec(-ff). Then G/N is isomorphic to a subgroup of
the symmetric group Sn, and so by Lagrange's theorem, \G/N\ divides
= n\.
1.3. Corollary. Let G be simple and contain a subgroup of index n 1.
Then \G\ divides n\.
Proof. The normal subgroup N of the previous corollary is contained in iJ,
and hence it is proper in G because n 1. Since G is simple, T V = 1, and
thus |G| = |G/JV| divides n\.
In order to pursue our main goal, which is counting, we need to discuss
the "orbits" of an action. Suppose that G acts on O, and let a G ft. The
set Oa {a*g \ g G G} is called the orbit of a under the given action. It is
routine to check that if (3 G 0
, then Op = Oa, and it follows that distinct
orbits are actually disjoint. Also, since every point is in at least one orbit,
it follows that the orbits of the action of G on ft partition ft. In particular,
if ft is finite, we see that \ft\ $^ |C?|, where in this sum, O runs over the
full set of G-orbits on ft.
We mention some examples of orbits and orbit decompositions. First, if
H C G is a subgroup, we can let H act on G by right multiplication. It is
easy to see that the orbits of this action are exactly the left cosets of H in
G. (We leave to the reader the problem of realizing the right cosets of H
in G as the orbits of an appropriate action of H. But be careful: the rule
x*h = hx does not define an action.)
Perhaps it is more interesting to consider the conjugation action of G
on itself, where the orbits are exactly the conjugacy classes of G. The fact
Previous Page Next Page