有限可換群の、部分群の個数を求める計算法

Abstract

A finite Abelian group is decomposed into a direct product of subgroups of prime-power orders.These subgroups are commutative p-groups,known p-Sylow groups.Every commutative p-group S is decomposed into a direct product of cyclic subgroups C（pk1）×C（pk2）×…C（pkt）, by an abbreviated notation（k1, k2, …,kt）. Let H be a type（h1, h2, …, hs）subgroup of S . First, we make bases（a1, a2, …,as）such that a'i s（i= 1 … s）are elements of S of order phi. Then we prove that there are （pf（h1）－pf（h1－1））（pf（h2）－pf（h2－1）＋1）…（pf（hs）－pf（hs－1）＋s－1） basis. Similarly, there are （pg（h1）－pg（h1－1））（pg（h2）－pg（h2－1）＋1）…（pg（hs）－pg（hs－1）＋s－1） basis of H. We prove that the number of type（h1, h2, …, hs）subgroups of S is Πsi＝1（pf（hi）－pf（hi－1）＋i－1） ───────────── . Πsi＝1（pg（hi）－pg（hi－1）＋i－1） Then we have a theorem to compute those of a finite Abelian group.