JOUBNAL
OF ALGEBRA
6, 222-241
(1967)
Sylow
Intersections
and Fusion
J. L.
ALPERIN*
Department of Mathematics, University
of Chicago, Illinois
60637
Communicated by I. N. Herstein
Received July 12, 1966
1.
INTRODUCTION
It is common
in mathematics
for a subject
to have its local
and global
aspects;
such is the case in group
theory.
For
example,
the structure
and
embedding
of subgroups
of a group
G may be usefully
thought
of as part of
the local
structure
of G while
the normal
subgroups,
quotient
groups
and
conjugacy
classes are relevant
to the global
structure
of G. Furthermore,
the
connections
between
local
and global
structure
are very
important.
In the
study
of these relations,
the methods
of representation
theory
and transfer
are very
useful.
The
application
of these techniques
is often
based upon
results
concerning
the fusion
of elements.
(Recall
that
two
elements
of a
subgroup
H of a group
G are said to be ficsed if they are conjugate
in G but
not in H.) Indeed,
the formula
for induced
characters
clearly
illustrates
this
dependence.
However,
more pertinent
to the present
work,
and also indica-
tive
of this connection
with
fusion,
is the focal subgroup
theorem
[8]: if P
is a Sylow p-subgroup
of a group G then P n G’ is generated by all elements of
the form
a-lb,
where a and b are elements of P conjzgate
in G. Hence,
this
result,
an application
of transfer,
shows
that
the fusion
of elements
of P
determines
P n G’ and thus P/P n G’ which
is isomorphic
with
the largest
Abelian
p-quotient
group of G.
It is the purpose
of this paper to demonstrate
that the fusion
of elements
of a Sylow
subgroup
P is completely
determined
by the normalizers
of the
nonidentity
subgroups
of P. Therefore,
P/P n G’, a global
invariant
of G,
is completely
described
by the local structure
of G. A weak form of our main
result is as follows : if a and b are elements of a Sylow subgroup P of the group G
and a and b are conjugate in G, then there exist elements a, ,.
.., a,,, of P and
subgroups HI ,.
.., H,,
of P such that
a = a, , b = a,
and ai and U~+~ are
contained in Hi and cmjugate
in N(H,),
1 < i < m -
1. We shall strengthen
* This
research
was partially
supported
by
National
Science