Normal p-complement
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The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter
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Definition
Symbol-free definition
A subgroup of a group is said to be a normal p-complement if it is a normal subgroup and it has a Sylow subgroup (particularly a Sylow -subgroup) as a permutable complement.
A group that has a normal -complement is termed a p-nilpotent group.
Definition with symbols
A subgroup of a group
is said to be a normal
-complement if it satisfies the following equivalent conditions:
-
is normal in
and there is a Sylow
-subgroup
of
such that
and
is trivial.
-
is normal in
and for every Sylow
-subgroup
of
,
and
is trivial.
-
is a normal Hall subgroup of
whose order is relatively prime to
and whose index is a power of
. In other words,
is a normal
-Hall subgroup of
.
If contains a normal
-complement
, we say that
is a p-nilpotent group.
Facts
Normal -complements may not always exist.
A complete list of normal p-complement theorems is available at: