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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.
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Statement

Hands-on statement

We can find a finite group whose order is a power of 2, 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 k, we can find a finite group whose order is a power of 2, that contains a Klein four-subgroup but does not contain a k-subnormal Klein four-subgroup.

Statement in terms of the weak normal replacement condition

The single-element collection \mathcal{S} comprising the Klein four-group is not a Collection of groups satisfying a weak normal replacement condition (?) for the prime 2.

Related facts

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:

G := \langle a,x \mid a^8 = x^2 = 1, xax = a^{-1} \rangle.

Then:

  • The subgroup \langle a^4, x \rangle is a Klein four-group.
  • There is no normal Klein four-subgroup: Any subgroup contained in \langle a \rangle is cyclic, so it cannot be a Klein four-group. Thus, a normal Klein four-subgroup, if it exists, must contain some element outside \langle a \rangle. However, the sizes of the conjugacy classes of elements outside \langle a \rangle is 4 each, so any normal subgroup containing an element outside \langle a \rangle 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 2^{k+3}, as follows:

G := \langle a,x \mid a^{2^{k+2}} = x^2 = 1, xax = a^{-1} \rangle.

Then:

  • The subgroup \langle a^{2^{k+1}},x \rangle is a Klein four-group.
  • There is no k-subnormal Klein four-subgroup: Any subgroup contained in \langle a \rangle is cyclic, so it cannot be a Klein four-group. Thus, a k-subnormal Klein four-subgroup, if it exists, must contain some element outside \langle a \rangle. Since all elements outside \langle a \rangle are in the same automorphism class, we can assume that it contains x. However, the smallest k-subnormal subgroup containing x, computed by taking the normal closure k times, is a dihedral group of order eight given by \langle a^{2^k},x \rangle.