Elementary abelian-to-normal replacement fails for Klein four-group
This article discusses a failure of replacement, i.e., a situation where the analogue of a valid replacement theorem fails to hold under slightly modified conditions.
View other failures of replacement | View replacement theorems
Contents
Statement
Hands-on statement
We can find a finite group whose order is a power of , that contains a Klein four-group (?) as a subgroup(i.e., it contains an elementary abelian subgroup of order four) but does not contain any normal subgroup that is a Klein four-group.
In fact, the following stronger statement is true: for any , we can find a finite group whose order is a power of
, that contains a Klein four-subgroup but does not contain a
-subnormal Klein four-subgroup.
Statement in terms of the weak normal replacement condition
The single-element collection comprising the Klein four-group is not a Collection of groups satisfying a weak normal replacement condition (?) for the prime
.
Related facts
Similar facts
Opposite facts
- Elementary abelian-to-normal replacement theorem for prime-square order: The statement that fails for
is true for odd primes.
- Abelian-to-normal replacement theorem for prime-cube order
- Jonah-Konvisser elementary abelian-to-normal replacement theorem
- Jonah-Konvisser abelian-to-normal replacement theorem
Proof
Example of the dihedral group (for the weaker version)
Further information: dihedral group:D16, Subgroup structure of dihedral group:D16
Consider the dihedral group of order sixteen:
.
Then:
- The subgroup
is a Klein four-group.
- There is no normal Klein four-subgroup: Any subgroup contained in
is cyclic, so it cannot be a Klein four-group. Thus, a normal Klein four-subgroup, if it exists, must contain some element outside
. However, the sizes of the conjugacy classes of elements outside
is
each, so any normal subgroup containing an element outside
must have order strictly bigger than four.
Example of the dihedral group (for the stronger version)
Further information: Dihedral group, Subgroup structure of dihedral groups
We need to take a dihedral group of order , as follows:
.
Then:
- The subgroup
is a Klein four-group.
- There is no
-subnormal Klein four-subgroup: Any subgroup contained in
is cyclic, so it cannot be a Klein four-group. Thus, a
-subnormal Klein four-subgroup, if it exists, must contain some element outside
. Since all elements outside
are in the same automorphism class, we can assume that it contains
. However, the smallest
-subnormal subgroup containing
, computed by taking the normal closure
times, is a dihedral group of order eight given by
.