| Research Abstract |
The following results have been obtained.
1. Let A be a finite cyclic p-group, and let G be a finite p-group on which A acts. If the number of complements of G in the semi-direct product AG is not divisible by the greatest common divisor gcd(p|A|, |G|) of p|A|, where |A| is the order of A, and the order |G| of G, then G is a exceptional p-group, namely, a cyclic p-group, a dihedral, generalized quaternion, or semi-dihedral 2-group. (This result has been obtained in the joint work with Masafumi Murai.)
2. Let A be the direct product of two cyclic p-groups, and let G be a finite exceptional p-group on which A acts. The number of complements of G in AG is divisible by gcd(|A|, |G|) of |A| and |G|. (This result has been obtained in the joint work with Tsunenobu Asai and Takashi Niwasaki.)
3. Let A be the direct product of a cyclic p-group and a cyclic group of order p^2, and let G be a finite p-group on which A acts. The number of complements of G in AG is divisible by gcd(|A|, |G|), which is a generalization of a theorem of P. Hall.
4. For the number of permutation representations of a finite abelian p-group, its p-adic properties have been obtained. In the proof, the generating function has been applied.
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