El, j times. The assumption 156.
6n=1 and x,y are products of cycles of lengths less than
or equal pn and x,y own picycles for i=1,2,...,n.
So obviously x,y are not pncycles. However, if both
x,y are p cycles, then they are both type II pncycles.
Hence by theorem 1 they are conjugate in N(T). For
general case beyond the condition 16i.
is similar to what we are going to do below, hence, is
omitted.
Define an increasing sequence of subsets of n as
follows:
A = E x E x x E x E x x x= En2 x A1 xQ
2= E x Ex .x E x x x = En3 x A2 xn2
(10.11)
Ad= E x E x xExAdx x x x .x Q = End1 x x Ad d
nl= An1 xnxQx ... .x = An1 xnl
A =n
n n
We will define the permutation x*,y* using A. as
1
vectors in cycles of x* and y* of lengths less than p
in such a way that the corresponding cycles of x and x*
are of the same type and so are the corresponding cycles
of y and y*.
For each d=1,2,...,n2. If a p cycle of x induces
51
a type I pdcycle of x,then it induces exactly p type I
d d
p cycles of x (see lemma 3.1). That is, xl is divisible
by p. Let x = p.m (this m is the same as md in (9.10)
and (9.11)). Now the total number of vectors in p cycles
of x is
(10.12) pd Xd = 6d+lPd+ d d (by the definition (10.10)
of 6d')
= (p6d).pd + (6d+l1).pd+
dd d d
= p .x1 + p .x2
d x d d
=p :.(P.md ) + p.x2
= pd+l.m + p.x hence,
dd d+l
(10.13) pdx (pd).p = ((6d+l)mx) pd+1 0 (modulo
pd+l) likewise, for y we have,
dd d d+l
(10.14) pd y (p d = ((6a+ll)m ).Pd1 0 (modulo
d+l)
d d
(10.15) hence, x2 y2 P5d (mod p). Let
(10.16) d = ExEx ...xEx(nAd) xxox...xQ then
(ndl)times d times
(10.17) d= (p6d)
then the number of vectors contained in type 7Ipdcycles
d+l
of x less 0a will be a multiple of p +(by (10.13),
and the same for y by (10.14). Define
(10.18) = ExEx...xEx(A d+E)xxnxx ...xn then
(nd2)times (d+l)times
(10.19) ed[ = (d+l1).pd+
Note that both sets d and ed work simultaneously for
both x and y because they depend only on 6d's and Ad's
which are the same for both x and y. It is easy to see:
(i) d and ed are disjoint
(ii) a+1 Ad = i d
(iii) If we define Jn_ = (An_)XQn_ = AnAn then
ni n2
n = An = U Id d=l ed disjoint union.
n n d=l d=1 d
We will use the vectors in Td as vectors in some
d d
(p6d) typell p cycles of x* and y* and use (6a+llmdx) d
vectors of ed as other (6d+llmdx) type II pdcycles
of x*, then the last mdx vectors of 8d as mdx type I
pdcycles of x*. Likewise, (6d+l1md pd vectors of ed
will be used to form type II pdcycles of y* other than
the (p6d) mentioned above, and the remaining md.p vectors
of 8d will be used to form all type I p cycles of y*.
See figure 4.3
6d+l
d
dP
for type II p 
_ycles of x*
(P6d) ^.p
vectors for
type II p cycles of x*
vectors for type I pdcycles
of x*
 3^~~ ~~  '
d+l
p
6d+llm
md
d
Figure 4.3
Now write x,y as products of cycles in an order that
cycles of smaller lengths first and in cycles of the same
53
length, type II cycles first, namely the order is: fixed
2
points, type II cycles, type I pcycles, type II p cycles,
2
type I p cycles, and so on. This way we get a onetoone
correspondence between cycles of x and y preserving the
cycle length. Although this correspondence does not
preserve the type but for each d, the first (p6d) p cycles
of both x and y have been ordered that they are all type II.
By (10.15) the number of pdcycles which x,y are of
different type is a multiple of p namely, (m$ mdY).p.
For each d=l,2,...,n1, define xd and y* on U8d
Ad+A (let 8 = n) inductively as follows:
d+l d nl
x ,y fix all vectors in A On = E x(QA)xl both
S1 1 1 n2x 1
*
x and yl map E x[a,b] to E x[a,b+l] for all aEPA,
1 1 n2 n2
*
ben. Thus, xI yl act on 1l as products of (p61)
2 x =
type II pcycles. Let B = {2,3,...,62m} and C 2
A2B2x1;then IB2I = 621m IC2Xl = mlx and define
2 21 2 1 2
B2y,C2y similarly. Then, let xl map E n3x[a,b,c] to
En_3X[a,b,c+l] for all b,cEQ and all a. B2 and map
En_3x[a,b,c] to En3x[a,b+l,c] for all b,ce and all
as C2x. Thus, x is defined on A2 so that the number
of type I (type II) pcycles is the same as that of x.
*
The definition of yl on A2 is similar. Obviously xl
*
and yl are conjugate.
Inductively, assume x and y are defined on
d1 d1
d1U ed1 = AdAdl;both are products of p cycles
and are conjugate and Xdl contains as many type I
d+l
p cycles as x does; y contains as many type I
d1
54
pd+lcycles as y does (same for type II pdcycles).
*
We then define xd* and y on TdU =Ad A as follows:
On H= End x(Ad)x d, let zd be the permutation on
Qd of order p (see lemma 4.1). Define xd* and yd* the
same on 1d by sending vector Endlx[a,wd] to End1x
[a,w zd] for all acnAd; then it is easily seen that
both xd* and yd* act the same as products of (p6d) type
II p cycles.
For each ceAd+l{fl let W=End2X{c}E nd1 and let
A={w}xd i n2 which are defined the same as in lemma 4.1.
Then by this lemma, there exists zeS n2 which is a pdl
cycle on A (note thatlAl=pdl), and choose a vector vEA,
let r=pd as in lemma 4.2. Note also that the vector v,
the element z and the set A all depend on the number
cE Ad+{1}.
(i) If cEBdX(BdY. We defined both x* and y* on End2
xcxQd+1 = AxxQ as 72 defined in (10.2). Hence, both are
products of p type II p cycles.
(ii) If ceBdxBdY then define x* to be 2 as in (10.2)
and define y* to be z1, in (10.1) on the set Ax2xQ.
(iii) If ceBdYBdX then define x* to be zl, in (10.1)
and y* to be 2 in (10.2).
(iv) If ceAd+l1BdxBdY define both x* and y* as 1
(in (10.1).
Since different c gives different i.'s i=1,2,...
on different (also disjoint) sets A's. As c ranges over
the set Ad+l{l), the union of all A's equals 8d (which is
defined in (10.18)). We have:
55
(a) xd* contains as many type I/type II p cycles as
d
x does. yd* contains as many type I/type II p cycles
as y does.
(b) For each cEAd+l {}, on the set En d2 x{c}x +i =
d+L nd2 l d+1
Axgxs either xd* and yd* are the same (case (i)'(iv) above)
or either one of xd* or yd* acts as zi while the other
one acts as z2 defined in (10.1) and (10.2). For c of
the latter case let gd be defined as in lemma 4.2 part (2)
on the set End x{c}x d+l = Axnxn Then by part 2(b)(c)
r r
gC EN(< t vi i t *v*> ) or gC factors into product of
i=l i=l v d
r elements each lies in N() for some i. Let
c x Y
Sbe identity map if c belongs to BdnB or A d1{}
BdXBdy = Cd CdY. Hence,
H gSd
(Xd) cEAd+l{l} Yd
By induction xd* and yd* are defined;hence, let
n1 n1
x*= n xd* Y*= l Yd*;then x is conjugate to x* and
d=l d d=l
y is conjugate to y* also by proposition 3.4. x and x*
are conjugate, y and y* are conjugate. Hence, conjugating
elementsgl,g2,g3 can be found such that x91=x* with
gl N(T) (y*)93=y with g3EN(T) and (x*)92 = Y* where
g2 factors into product of elements each either comes
from members of F1 or members of F2. In fact, g2= II cg
The proof of theorem 2 is then completed.
11. Fusion in Direct Products
Since the Sylow psubgroups of the symmetric groups of
degree m, where m is any positive integer, are the direct
56
products of the groups Spn's for various n's as discussed
in the preceding sections, to know the fusion of elements
in such sylow subgroups, it suffices to know the fusion
of elements in the direct product of two S n's. In this
section we will discuss the local fusion of the direct
product of two sylow subgroups Spn's with the same degree
pn. For discussion of direct product of two Spn's with
different n's, the proof is similar, hence, is omitted.
Before going into direct product, let's look at a
single sylow group Spn for a moment. Recall each con
jugacy class of Spn, except the class containing long
pncycles, corresponds to a cycle decomposition, namely,
a ntuple of constants (nonnegative integers) {CC1,C2,C2...
,Cnl} satisfying the relation
CO + PC1 + p2C2 + ... + Pn1Cn = n
where for each d=0,1,2,...,nl. Cd is the number of
pdcycles in the cycle decomposition. Note that CO+pC +
p2C2++P d = 6 d+ *d+l as defined in (10.10) which is
d+l
the number of vectors in cycles of lengths less than p
Again, we assume 16i.
the increasing sequence of sets A,A2,...,An as in (10.11).
Then we define the permutation x* using A. as set of
1
vectors in cycles of x* of lengths less than pi as we
did in section 10 with the choice of the element x in
the conjugacy class such that all cycles of x are type II
(except pocycles). Thus all cycles of x* are type II
except pcycles and x* is uniquely determined by the
57
conjugacy class. We call this unique element x* of the
conjugacy class the standard element of this conjugacy
class. The example below shows how the standard element
of a conjugacy class is found:
EXAMPLE 4.3: Let p=3, n=4 the conjugacy class cor
responds to the constants C0=12,C ==2, C2=4, C3=1, contains
the standard element x* =
([1111)1112[(1113)([1121)([1122 ]Y[1123)([1131][1132][1133
([1211])([1212]X[1213]1[l221] [1222] [1223])61231] [1232][1233])
([1311][1321][1331] [1312][1322][1332] [1313][1323][1333)
([2111] [2121] [2131] [2112] [2122] [2132] [2113] [2123] [2133])
([2211][2221][2231] [2212][2222][2232] [2213][2223][2233)
([2311] [2321] [2331] [2312] [2322] [2332] [2313] [2323] [2333)
([3111] [3211] [3311] [3121] [3221] [3321] [3131] [3231] [3331]
[3112] [3212] [3312] [3122] [3222] [3322] [3132] [3232] [3332]
[3113][3213] [3313] [3123] [32231. [3323] [3133] [3233] [3333])
Notice that this example goes beyond the condition
1<6.
 1
substantial and hence the standard element can be found
in general.
Now let S and S' be Sylow psubgroups of the sym
metric groups ypn and ypn' acting on the set on and n '
respectively. Consider the group)= S x S', it is easy
to see, is a Sylow psubgroup of y2.pn acting on the set
nU gn' (disjoint union),and it is also the Sylow psubgroup
of the subgroup y nx Ypn' of Y2.pn where pnx Ypn' is
considered as the subgroup which "stabilizes" both sets
n and on'; that is, the permutations which maps elements
of Qn into n elements of n into n '.
Let the notations (5.6)(5.10), (6.1)(6,7), (9.1)
and (9.2) be defined on S as well as on S' with the lash
"'" added to all those in S'. Consequently, we have
notations for "products", namely, we have Ai x A',
i=0,1,...nl, in particular T x T' = the group generated
by all tv(nl)'s and t (n1)' s. Also, besides F1,F2,
F 1'F2 we may define
(11.1) F* = {N(TxT')}UF1UF'
(11.2) F* = {N(TxT')}UF2UF2'
Now let xx' and yy' be two conjugate elements of
S* s*
= S x S' where x,yeS and x',y'eS'. Let x*, x y*, y
be the standard elements of the conjugacy classes containing
x,x',y,y' respectively; then, by the main theorem, there
exist gl'g 9g2 2 which factor into products of elements
in the family F ,F1',F1,Fl' respectively (or F2,F2',F2,F2'
respectively) such that
x91 = x* (x') g = x
92 9
Y 2 y (y' g2
y = y* (y') =y
hence
(11.3) (xx')191 = ((xx')l = (x l.x91 g1
= (*.')1 = (x*)91 x = x*.xI
Here the qualities hold because gl centralizes x' and
gl' centralizes x4, likewise
(11.4) (yy')292' = y*y *
Thus F (and F *, too) control the local fusion from
xx' to x*x* and from yy' to y*y' Hence, the last
1* *
problem is how the element y*y and x*x' are fused together.
We claim in the following lemma that the normalizer (N(TxT'))
will give the local fusion from x*x* to y*y'* (that is
the reason why we throw N(TxT ) into both definitions
of F* and F*):
1 2
LEMMA 4.4 Let x,yeS, x',y ES', if xx',yy'E=
SxS' are conjugate and if x*,x'*,y*,y'* are standard
elements of the conjugacy classes containing x,x',y,y'
respectively, then there is gEN(TxT') such that
(x*x'*) =y*y,*.
PROOF Since we will be dealing with the four elements
x*,x'*,y*,y' through the proof, to avoid writing the
"*" all the time, we will use x,x',y,y' to denote x*,x'*
y*,y' That is, we assume x,x',y,y' are standard elements
themselves such that xx' is conjugate to yy'.
Let Cx, C ...,C C...,C
',C f' .., C x'y o 0 !1' n11
x' '...n } { ,C ...,Cn } be the constants
corresponding to these conjugate classes. We have, because
xx' is conjugate to yy', that for each i=0,1,2,...,nl
(11.5) (CxC)=CxCy
i 1 i i
For i=0, we have CYCX=CX'CY; let V be the set of
O O O O O
vectors in which are fixed by exactly one of the two
elements xx' and yy', and let V' be the set of vectors
0
in n which are fixed by exactly one of the two elements
xx' and yy'. By the way the standard elements are defined,
it is not too difficult to see that I Vo = V' =
CYCxI=IcX'CY'I=f We call vectors in V and V'
extra" vectors of pcycles of xx and yy'.
"extra" vectors of pocycles of xx' and yy'.
60
They form "blocks"{v.xv} and{w'x' } for some v and w' in
n1 (recalling all CoX,Co,C Y,C Y' are multiples of p.
Hence, f is also a multiple of p).
Now let gl be the permutation, which acts on VOUVO'
and leaves all other vectors fixed, defined by sending
blocks onto blocks,and in each pair of corresponding
blocks vxst and w'xo' it maps [v,a] [w',a] all aen
Then, (xx')1 and yy' have the same set of fixed vectors
and obviously geN(TxT') because if g interchanges the
blocks vxQ and w'xQ' then (tv)g = tw' and (twI') = tV.
We have used gl as conjugating element to xx' to pass
from xx' to the element (xx')gl which is also a product
of standard elements and has the same fixed vectors as
fixed vectors of yy' in the sense that they are closer
"looks."
Inductively, we find elements gl,g92,...gd all in
N(TxT') such that (xx')9192,'**9d and yy' have the same
vectors in all cycles of lengths less than pd;then let
Vd be the set of vectors in Q which lie in one and only
one pdcycle of either (xx')gl ''gd or yy', and let
Vd' be defined likewise. Then, define gd+l to be the
permutation acting on VdUVd' leaving all other vectors
fixed by interchanging vectors in pdcycles of (xx')gl''''gd
and yy'rthus making the elements (xx')g91'*gd+l and yy'
have the same vectors in all cycles of lengths less than
d+l
pd. By induction, we have the elements gl,g2,...gn all
in N(TxT'). Hence, g=g1g2...gne N(TxT') such that (xx')9=
yy'. Conclude the proof of the lemma.
61
Let's look at an example which illustrates the proof
of the lemma;
EXAMPLE 4.5 Let p=3, n=3, in S27 x S2/. The standard
elements x, x', y, y' are given by the constants:
d Cd Cd' CdY CdY'
0 3 9 6 6
1 2 3 4 1
2 2 1 1 2
We first write the cycle decomposition of xx' and yy'
then underline the cycles of xx' which are contained in
yy', and find the sets V0 and Vo' hence the conjugating
element gl. Secondly, write the cycles of (xx')gl and
underline the cycles of (xx')91, which are contained
in yy'. Hence, find the sets V1 and VI' and then the
conjugating element g2. In this example, the sets V2 and
V2' are empty;hence, (xx')1l92 = yy'. The local conjuga
tion takes two steps to go from xx' to yy'. See the
following page.
n en en en
Nl ri n e
'
N en N n
in N en N
N en en e
N mI N m
en N en N
N N'
eN IN
N M N N
N en:N en
N inm N e
I _
NM
r r^, ru n
N rI N n
r i r< rM
N e IN en
r L
e iN N
r I I'
N en N e
r N  m rs m
N N N
n N N
en N en N
N m o N
m CM H m M
il m Ii4 rs
04 04 04 rM
en Nen N
V i" r 04'
en( eNen N
m r( P i m i
'en
e nIN en
N enHI  en
en N; N
r ' m < 24
N N ins~
ni~N en( N
.1N en en
en N en en
i  04 il
i i m N rl
iit i r 1 
rsc~, 04040
ri0 ( 04
r4 u r 4 m
i I04 r ^
r1 .c m 
i 
x 04 r
N
m N
N mn
M m
NM m
N eN
N eN
en 
? e
N m
N N
C m
N rN
en N
e4 N
O 0
N I
r^ i
i r
6 N
N en
en e : 
N N N
en N

en e
N IN
eni I
0'1
^l m
O ^
m ri
r
r^
n rl"
m c~
r
^ ^_ ^_
f i C
N I(
1 m c
r<+ ml
rc t sim
r( rti r
N'f ~ ryl
ri m
IN'*s..
rs V* *m m r
i rO r\ m
. i r rM
II II
0 0r^ r
> > I
n N
0N m
3 0/
0N m
N en
en N
Ni m
N 0e
Ne m
NeN
en N
N N
en en
en en
N e
n N
e Nn N
04 04
N m
e N
m: 1?1

N
X4m
2
N N
m r?
m mi
04 m
nm 0
r i N
m r
 N
N N N en
N N
N N en N
 en NN
m ~m
rri r^
en,.en en4N
N N en en
e N~
m ' H* en N 
( N
r cN csi 4 f
Ntr, .. in 
NNi
0 0' 
m04' r
x
rl r< *'l" 
04 0 0 
^ ^ S^
(M 0
CM,, 04 p
r * rri 0
(M ^ ?& ^^
04 N +0 4 r
Fs 0 0 0
rl^ ^5. *
CM, I* M ft n
(M r) 0 ~
i ^ / 
f .5.'^
ir **11 m t
04 oi r> r
x tt 0
63
As a corollary of the main theorem and this lemma,
we have:
CONCLUSION: In the Sylow psubgroups of a symmetric
group of any degree, the local fusion of elements is
given by the normalizers of groups of only two different
structures; one is a big elementary abelian group of ppower
rank, the other is the small elementary abelian group of
order p.
APPENDIX
In this appendix we will see four examples of permu
tations x, x*, y, y* in Spn with p=5, n=3 and x, x*, y, y*
are all conjugate in Ypn such that x is conjugate to
x*. These examples illustrate: (1) how the conjugating
element glEN(T) is found to make xg1=x*; (2) how the
element g2,EN(T) is found to make (y*)93=y; (3) how the
element g2 is found so that (x*)92=y* and g2 factors into
product of elements in the normalizers of the family of
psubgroups of the form with v(n2)
ranges overthe set _n2 mentioned in the main theorem.
In fact, how the elements x* and y* can be found
from x,y as mentioned in Theorem 2 can also be seen im
plicitly from these examples.
Each example gives the element first by the maps
x0,xl,x2,...Xn_1 as defined in (5.9), then by its cycle
decomposition. All cycle decompositions are written in
the same manner such that the corresponding cycles all
have the same length and/or the same type. The type
of each cycle (or cycles) is put on the top of the cycle
(or cycles). The elements gi are defined to map the vectors
into the vectors in the corresponding positions of the other
element. The illustration of how gleN(T) and how g2 factors
65
into product of elements in N()
can be seen to some extent.
1 (
3
o X
*' o ' 
U I
N N. .
in l .
N 
0
II >
a x
E
0
441
s ~ N 
in N '5 '
o i i '
N in 4
r, N N i
N in iN N
N in '5 N
N '5 in N '
N in t! i i
N Lfl 5 4
N i N 
> 
'5 Vin inin
N '5 '5 5 '
N i '5 4
N '5 '5 N N
Cl '5 5 N '
N n '
Hi~n N? MNI MN NV
.4 HC H ~ LN
N '5 '5 4 
mN in 4 N
in N 5 i
4 N N '5 '
N n 5 H
'5 i ~9n 4 4
,4 N N N N
MN mN inM,4
NI in '5n Hr HN
4( m 4? NO in
N in ,4
in '5 N N
N 4 LN 5 
in in O ,H
'5 N in n '
04 N
Cin i '5 iH
N Hn MNMi N
N H 4 H i i
N N '5 i
in in N
,4 4 N N
N 4,NI MN M's
N Nr UN N, N
N N '5 N iN
N N ?
II
II
11H
x
II O
X 
a
0
m *
r=  ~ ~
in in in in in
in in in in in
,4 N N '5 in
'5. '5 '5 in in
in in in '5y '
( N N 5 in
N N N in in
in in in N N
4 N N in
Nj N N in
in in in N^ N
N N '3 in
m m m a 
in in in 4 H
r4 N N 'm in
in in in '5 '
'5r '5 in in
4 N N '3 in
N MN M M'
s H'S H's N N
f4 NM N^ in
N HN N '
5 Nr 5 N N
1 N N '5 in
N M V *t i
4 4 '5
4 N N '5 i
in i in N
N1" N^ N in in
r4 N N '5 in
N fN N^ '5" '
MN P MNf' MN~ MN^ MNr
MNp MN MN^ N MN
(N M fN N~ N^
N fN N N
i4 N^ N in
^4 rN N in
~n l ~a
in in i N (N
M rN N in in
H NM '5 in
H N4 Ni in
rN MN MN MNs (N
NM M N MN Nj N
N NM '5^ in
,4 Hf 41 N N
rN N N H
4i fN Ns '5 in
rin in; in 4
4 NM N '51 in
00< '
,4 N N '5 in
H" N N '5 in
N3 Ny N ,4 r4
r [N N '5 in
,4( N N '5T in
L ~ ( 
11 (M = m t i
Io
.0
4i
If
4
oN c o
1 .01
S ,0 O
N l
O
mI II
II *O
'a,
E
x
U)!
) U) U) U) in
n) in Un W) U)
4 IN CoInU
Co Cm U) U U
w Hn m 14
14 N! C r U)
U N N N
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REFERENCES
[1] Alperin, J. L., Sylow Intersections and Fusion,
J. Alg. 6 (1967) 222241.
[2] Alperin, J. L., Up and Down Fusion, J. Alg. 28
(1974) 206209.
[3] Alperin, J. L., Finite Groups Viewed Locally,
Bul. AMS 83 No.6 (1977). 12711285.
[4] Alperin, J. L. and Gorenstein, D., Transfer and
Fusion in Finite Groups, J. Alg. 6 (1967) 242255.
[5] Dolan, S. W.,Local Conjugation in Finite Groups,
J. Alg. 43 (1976) 506516.
[6] Finkel, D.,Local Control and Factorization of
Focal Subgroups, Pacific J. Math. 45 No.l
(1973) 113128.
[7] Glauberman, G., Global and Local Properties of
Finite Groups, Academic Press, New York, 1971.
[8] Goldschmidt, D. M., A Conjugation Family for Finite
Groups, J. Alg.16 (1970) 138142.
[9] Gorenstein, D., Finite Groups, Harper & Row,
New York, 1968.
[10] Hall, M., The Theory of Groups, Macmillan, New York,
1959.
[11] Herstein, I. N., Topics in Algebra, John Wiley & Son
Inc.,New York, 1975.
[12] Higman, D. G., Focal Series in Finite Groups,
Canadian J. Math. 5 (1953) 477497.
[13] Huppert, B., Endliche Gruppen I, SpringerVerlag,
New York, 1967.
[14] Kaloujnine, L.,La Structure des pgroupes de Sylow
des Groupes Symetriques Fins, Ann. Sci. Ecole
Norm, Supor 65 (1948) 239276.
[15] Kaloujnine, L., Sur les pgroupes de Sylow du Groupe
Symetriques du Degree' m, C. R. Acad. Sci. Paris
221 (1945) 222224. P
[16] Kantor, W. M. and Seitz, G. M., StepbyStep Conjugation
of psubgroups of a Group, J. Alg. 16 (1970)
298310.
[17] Passman, D., Permutation Groups, Benjamin, New York,
1968.
[18] Rotman, J., The Theory of Groups; An Introduction,
Allyn & Bacon, Boston 1968.
[19] Schreider, 0., Uber die Erweiterung von Gruppen I,
Monatsh Math. Phys. 34 (1926) 165180.
[20] Thompson, J. G., Normal pComplements for Finite
Groups, J. Alg. 1 (1964) 4346.
[21] Weir, A. J., The Sylow Subgroups of the Symmetric
Groups, Proc. AMS 6 (1955) 534541.
BIOGRAPHICAL SKETCH
Johntien Hsieh was born 1946 at Tainan City of Taiwan,
Republic of China. He studied mathematics at National
Taiwan Normal University in Taipei during the years
19651970 and got his B.S. degree in 1970. He came to
University of Florida as a graduate teaching assistant
in the fall of 1973 to study mathematics and got his
M.S. degree in the summer of 1974.
Mr. Hsieh is married to Lienzu Lin. They have a
daughter named Ellen who is now 6 years old. Mrs. Hsieh
is also a mathematics major graduate student studying
at the University of Florida.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
David A. Drake Chairman
Associate Professor of
Mathematics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Mark P. Hale Jr. Cochairman
Associate Professor of
Mathematics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality
as a dissertation for the degree of Doctor of Philosophy.
James K. Brooks (,
Professor of Mathematics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Thomas E. Bullock
Professor of Electric
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality
as a dissertation for the degree of Doctor of Philosophy.
Katherine B. Farmer
Assistant Professor of
Mathematics
This dissertation was submitted to the Graduate Faculty
of the Department of Mathematics in the College of
Liberal Arts and Science and to the Graduate Council,
and was accepted as partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
December 1978
Dean, Graduate School
UNIVERSITY OF FLORIDA
3 1262 08554 0796