Group of prime power order may have multiple characteristic subgroups of prime order

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Statement

Let p be a prime number. Then, there exists a group of prime power order P that has more than one characteristic subgroup of order p.

Related facts

Proof

Case p = 2

Further information: SmallGroup(16,4), subgroup structure of SmallGroup(16,4)

For p = 2, we can take:

P := \langle a,b,c \mid a^2 = b^4 = e, b^2 = c^2, ab = ba, ac = ca, cbc^{-1} = ab \rangle.

Then, P has three characteristic subgroups of order two:

  1. The subgroup \langle a \rangle is the commutator subgroup.
  2. The subgroup \langle b^2 \rangle is the unique subgroup generated by the unique element that is a square but not a commutator.
  3. The subgroup \langle b^2a \rangle is the unique subgrou pgenerated by the element that is a producto f squraes but not a square itself.