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69
III APPLICATIONS
by
A. N. Milgram
A. Solvable Groups.
Before proceeding with the applications we must discuss certain
questions in the theory of groups. We shall assume several simple propo
sitions: (a) If N is a normal subgroup of the group G, then the mapping
f(x) = xN is a homomorphism of G on the factor group G/N. f is called
the natural homomorphism. (b) The image and the inverse image of a
normal subgroup under a homomorphism is a normal subgroup, (c) If f
is a homomorphism of the group G on G
1
, then setting N
f
= f(N), and
defining the mapping gasg(xN) = f(x)N',w e readily see that g is
a homomorphism of the factor group G/N on the factor group G'/N
1
.
Indeed, if N is the inverse image of N' then g is an isomorphism.
We now prove
THEOREM 1. (Zassenhaus). If U and V are subgroups of G, u and
v normal subgroups of U and V, respectively, then the following three
factor groups are isomorphic: u(UnV)/u(Unv),
v(UnV)/v(unV), (UnV)/(unV)(vnU).
It is obvious that U n v is a normal subgroup of U n V. Let f
be the natural mapping of U on U/u. Call f(UnV) = H and f(Unv) = K.
Then f'^H) = u(UnV) and f^K) = u(Unv) from which it follows
that u(UnV)/u(Unv) is isomorphic to H/K. If, however, we view f as
defined only over U n V, then f^K) = [un(UnV)](Unv) =
(unV)(Unv) so that (UnV)/(unV)(Unv) is also isomorphic to H/K.
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Thus the first and third of the above factor groups are isomorphic to
each other. Similarly, the second and third factor groups are isomorphic.
Corollary 1. If H is a subgroup and N a normal subgroup of the
group G, then H/HnN is isomorphic to HN/N, a subgroup of G/N.
Proof: Set G = U, N = u, H = V and the identity 1 = v in
Theorem 1.
Corollary
2.
Under the conditions of Corollary 1, if G/N is
abelian, so also is H/HnN.
Let us call a group G solvable if it contains a sequence of sub
groups G = G
0
D Gj D. . .D G
g
= 1, each a normal subgroup of the
preceding, and with G^ /G
i
abelian.
THEOREM 2. Any subgroup of a solvable group is solvable. For
let H be a subgroup of G, and call H
£
= HnG.
. Then that H.^/H. is
abelian follows from Corollary 2 above, where G
M
, G. and H. j play
the role of G, N and H.
THEOREM 3. The homomorph of a solvable group is solvable.
Let f(G) = G
1
, and define
G\
= fCG^ where G. belongs to a
a sequence exhibiting the solvability of G. Then by (c) there exists a
homomorphism mapping
G
il
/G
i
on GJ^/GJ. But the homomorphic image
of an abelian group is abelian so that the groups G'. exhibit the
solvability of G' »
B. Permutation Groups.
Any one to one mapping of a set of n objects on itself is called
a permutation. The iteration of two such mapping is called their product.
71
It may be readily verified that the set of all such mappings forms a
group in which the unit is the identity map. The group is called the
symmetric group on n letters.
Let us for simplicity denote the set of n objects by the numbers
1,2, .
.. , n. The mapping S such that S(i) = i + l mod n will be de
noted by (123.
. .n) and more generally (i j. .
. m ) will denote the map
ping T such that T(i) = j, . .
. ,T(m) = i. If(ij.
..m ) has k numbers,
then it will be called a k cycle. It is clear that ifT = (ij.
..s) then
T
1
=(s.
..ji>.
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This note was uploaded on 11/14/2011 for the course MATH 367 taught by Professor Sdd during the Spring '11 term at Middle East Technical University.
 Spring '11
 sdd

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