Formula for number of maximal subgroups of group of prime power order
In particular, for , this number is congruent to 1 mod .
- Equivalence of definitions of maximal subgroup of group of prime power order
- Fourth isomorphism theorem
- Equivalence of definitions of size of projective space
The key idea behind the proof is that the maximal subgroups of correspond, via the fourth isomorphism theorem, to maximal subgroups of the Frattini quotient . The latter is an elementary abelian group of order and the number of maximal subgroups equals the number of codimension one subspaces, which is the indicated value by Fact (3).