有限可換群の、部分群の個数を求める計算法

概要

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A finite Abelian group is decomposed into a direct product of subgroups of prime-powerorders.These subgroups are commutative p-groups,known p-Sylow groups.Everycommutative p-group S is decomposed into a direct product of cyclic subgroupsC（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 elementsof 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 isΠsi＝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.
都留文科大学の論文
1999-10-00