Sylow theorems
Three theorems on maximal -subgroups in a finite group, proved by L. Sylow [1] and playing a major role in the theory of finite groups. Sometimes the union of all three theorems is called Sylow's theorem.
Let be a finite group of order
, where
is a prime number not dividing
. Then the following theorems hold.
Sylow's first theorem: contains subgroups of order
for all
; moreover, each subgroup of order
is a normal subgroup in at least one subgroup of order
. This theorem implies, in particular, the following important results: there is in
a Sylow subgroup of order
; any
-subgroup of
is contained in some Sylow
-subgroup of order
; the index of a Sylow
-subgroup is not divisible by
; if
is a group of order
, then any of its proper subgroups is contained in some maximal subgroup of order
and all maximal subgroups of
are normal.
Sylow's second theorem: All Sylow -subgroups of a finite group are conjugate.
For infinite groups the analogous result is, in general, false.
Sylow's third theorem: The number of Sylow -subgroups of a finite group divides the order of the group and is congruent to one modulo
.
For arbitrary sets of prime numbers, analogous theorems have been obtained only for finite solvable groups (see Hall subgroup). For non-solvable groups the situation is different. For example, in the alternating group
of degree 5, for
there is a Sylow
-subgroup
of order 6 whose index is divisible by a number from
. In addition, in
there is a Sylow
-subgroup isomorphic to
and not conjugate with
. The number of Sylow
-subgroups in
does not divide the order of
.
References
[1] | L. Sylow, "Théorèmes sur les groupes de substitutions" Math. Ann. , 5 (1872) pp. 584–594 |
[2] | M. Hall, "Group theory" , Macmillan (1959) |
Sylow theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sylow_theorems&oldid=32056