[過去ﾛｸﾞ] ガロア第一論文及びその関連の資料スレ (1002ﾚｽ)
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111(1): 2023/01/31(火)15:56 ID:tkHk7/Du(4/5) AAS
>>110
つづき

In both the cases, the group of the equation can be partitioned by adjunction into groups such that one can pass from one to another by a self-transformation;but the condition that these groups have the same substitutions holds only in thesecond case.
This is called proper decomposition.
In other words, when a group G contains another, H, the group G can be parti-tioned into groups each of which is obtained by operating on the permutations inH a self-transformation, in such a way that,G = H + H S + H S' + ・・..And we can also partition into groups which have all similar substitutions, such that G = H +TH +T'H +・・
These two types of decompositions generally do not coincide. When they do coin-cide the decomposition is said to be proper.

It is easy to see that, when the group of an equation is not susceptible to any proper decomposition, however well we might have transformed this equation, the groupsof the transformed equations will always have the same number of permutations.On the contrary, when the group of an equation is susceptible to a proper decom-position in such a way that we can decompose it into M groups of N permutations,we can resolve the given equation by means of two equations: one will have a groupof M permutations and the other, one of N permutations.
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112: 2023/01/31(火)15:57 ID:tkHk7/Du(5/5) AAS
>>111
つづき

Hence when we would have exhausted all possible proper decompositions on the group of an equation, we arrive at groups which can be transformed but for whichthe number of permutations will always be the same.
If each of these groups has a prime number of permutations then the equation will be solvable by radicals; otherwise, not.
The smallest number of permutations that an indecomposable group can have,when this number is not a prime number, is 5・4・3.（＝60のA5(交代群)）
(引用終り)
以上

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