Congruence condition fails for number of characteristic subgroups in group of prime power order
Let be any prime number.
It is possible to have the following situation: is a finite p-group of order , and there exists with such that the number of Characteristic subgroup (?)s (see also Characteristic subgroup of group of prime power order (?)) of of order is a nonzero number that is not congruent to 1 mod .
- Congruence condition fails for number of characteristic subgroups in abelian group of prime power order
- Congruence condition on number of subgroups of given prime power order tells us that the opposite is true if we are looking at all subgroups or at all normal subgroups.
- Congruence condition fails for number of normal subgroups of given prime power order: Note that this states that the number of normal subgroups of a given prime power order may be a nonzero number not congruent to 1 modulo the prime, but to construct a counterexample, we need to move to an ambient finite group that is not itself a p-group.
Further information: central product of D8 and Z8
Consider to be the central product of D8 and Z8, a group of order 32 (GAP ID: (32,38)). has two characteristic subgroups of order 8: