の-Subgroup of finite nilpotent Groups
ShSsaburo FUJITA
(Ｓｅ･。山la･ｒ ｏｆ Ｍａtｋｅ。ｌａti。'、Ｆａｃｉｄりof Litｅｒａtｕｒｅ ｉｕｉｄ Ｓｃｉｅ、ｌｃｅ、Ｋｏｃｈｉ ｕni、ｅｒｓｔり)
In this note, we shall study some properties of the I/'-subgroups of finite nilpotent groups･
ＮＯＴ八TIONS : Ｇ denotes ａ group of finite order. ψ(Ｇ)，￡)(Ｇ)ａｎｄZ (G) denote the ψ-subgroup, the commutator subgroup and the centre of ａ group Ｇ respectively.φｓ(Ｇ) denotes the intersection of ａ group Ｇ with its maximal subgroups containing Ｎ. Theorem １. The 0 -subgroup of a finite group G is the intersection of normal subgroups ｙ of G such that the φ･subgroup of the factor group Ｇ/Ｎ is ａ unit subgroup of G/N.
Proof. It is evident that 0n(.G) containｓ Ｎ ａnd 0iG). Since Ｎ/Ｎ＝φ(ＧＩＮ)⊇
Ｎφ(Ｇ)IN ａnd also φ(GZΦ(Ｇ))＝φ(Ｇ)/ゆ(Ｇ)，ｗｅ get ｎＮ＝φ(Ｇ).
Corollary 1. Let Ｇ be nilpotent, then the factor group G/N is the direct pro-duct of groups of prime order if and only if ｙ contains the φ-subgroup of G.
Proof. Since Ｇ is nilpotent, Ｇμ〉(G) is abelian and ψ(GZΦ(Ｇ))＝φ(Ｇ)/ψ(Ｇ). Hence ＧＺの(G) is the direct product of groups of prime order''. If N contains 0(G), then, from Ｇ/ｊＶａＧ/ψ(Ｇ)/ｙ/の(G), it follows that Ｇ/Ｎ "iｓthe direct product of groups of prime order.
Conversely, if G/N is the direct product of groups of prime ordec, thenの(Ｇ/Ｎ) = N/N and therefore jｖ２ψ(Ｇ)fｒｏｍ Theorem １.
Theorem 2. If ａ normal subgroup ｙ of Ｇ is contained in the 0 -subgroup and the factor group G/Ｎ is cyclic, then Ｇ is cyclic.
Proof. We put ＧＩＮ= [aN], then we have G={a]N. Since iV￡Φ(G), follows G={a].
Theorem ３.Ｈ ａ normal subgroup N of G is contained in the centre and the factor group G/N is nilpotent, then Ｇ is nilpotent.
Proof. If ｙ is contained in 0{G), then, by assumption, 0(Ｇ/Ｎ)＝の(Ｇ)ZN ， contains￡)(Ｇ/Ｎ)＝ＮＤ(Ｇ)/Ｎ ａnd so ゆ(Ｇ)三￡)(G). Hence Ｇ is nilpotent. If ｙ is not contained in 0(G), then Ｇ has ａ maximal subgroup G＊1 such that G = G*iN and so Ｇ＊1 is normal in Ｇ since ｊＶΞＺ(Ｇ).Ｆｒｏｍ G/G*i=N/G*,nN, it follows that Ｇ/Ｇ＊1 is abelian and so we get Ｇ＊1ユD(G). Let Gi be ａ maximal subgroup of Ｇ containinｇ Ｎ. Since GJN3.0(Gil＼f)⊇￡)(ＧＩＮ), we have Ｇ１ユ￡)(Ｇ).Thｕs we find that￡)(Ｇ)is con-tained in the intersection of all maximal subgroups,, that is, ￡)(Ｇ)旦ψ(Ｇ)ａｎd hence Ｇ
40 J知大学学術研究報告第９巻 自然科学 Ｉ 第５号 is nilpotent.
Corollary ２. Ｈ ａ normal subgroup･ｙ of G is contained in the centre, the factor group G/N is abelian and 0(.G)nN=[, then Ｇ is abelian.
Proof. From Theorem ３，G is nilpotent and so ゆ(Ｇ)ユＤ(Ｇ).ＡｓＧ/Ｎ'is abelian, 7Vユ￡)(G). Thus￡)(Ｇ)辰砂(Ｇ)ｎＭthat is, ￡)(Ｇ)＝1.
Theorem ４. Let G be nilpotent and H its subgroup, 山ｅｎψ(Ｇ)こゆ(と)．
Proof. Let １⊂Ｈ=Ｈ,⊂H2⊂…‥⊂Hk-i⊂j飛＝Ｇ be ａ normal chain. Ｔｈｅｎψ(仔)
is norma卜n Hui and so ゆ(昂)辰の(昂。i), 7:= 1,2,｡‥‥‥，た−1.2)Thｕs we get ψ(Ｇ) ユの(耳).
Theorem ５. If G is nil potent and Ｎ 1ts normal subgroup, thenのn(G) = N0(G). ＰＲＯＯＩ・. If N is contained in の(G), thenのｗ(Ｇ)＝Ｎφ(G) holds clearly. If N is not contained in 0(G). then Ｇ has ａ possible minimal subgroup H such as Ｇ = NH. Let G, be ａ maximal subgroup of Ｇ containing Ｎ， then C = GiH. From Ｇ／Ｇｌ１召/ＧＩｎＨ， it follows that Hi = G,r＼H is ａ maximal subgroup of Ｈ and Ｇ.,= ＮＨ，.Conversely, Ｇ＊,＝ＮＨ＊1 wheｔｅ Ｈ＊l is an arbitrary maximal subgroup of H, is ａ maximal subgroup of Ｇ containing Ｎ. As the maximality of NH, and Ｇ＊l are showed in the same way, we shall show about G*,. If ｙｎ尽⊃NOH*,, then (H*,･．ＮｎＨ＊１)＝(μ:Nf]H) and sｏ(Ｇ＊1 :ｙ)＝(Ｇ‘:N). This conflicts with G＊1⊂Ｇ and hence Nr＼H=Nnだ*,. Then (/-/･．ＮｎＨ): (Ｈ＊1:ｊＶｎＨ＊1)＝(Ｇ:Ｎ):(Ｇ＊1:AO, and therefore Ｇ＊1 is ａ maximal subgroup of Ｇ. Thus we have 0n(G) = Nの(耳)。On the other hand, it is evident that のn(G)ユN0(G) and by Theorem ４，の(Ｇ)ユ0(.H) holds. Thus we getφ。(Ｇ)＝Ｎの(G).
Corollary ３. If there is ａ homomorphism of Ｇ onto Ｇ' and Ｇ is nil potent- then there is ａ homomorphism of <D{G) onto 0(Gり.
Proof. Let Khe the kernel of the homomorphism of G onto Ｇ≒then Ｇ∼ＧノＫ三 C. From Theorem ５，it follows that ψ(Ｇ/Ｋ)＝Ｋφ(Ｇ)/Ｋａψ(Ｇ)/尺∩の(G) and there- fore 0(G)∼ψ(Ｇ/Ｋ)ａの【Ｇり.
Theorem ６. Let Ｇ be ｎｉ】potentand N its normal subgroupバfの(Ｇ)ｎｙ＝の(yＶ)， 山ｅｎ the maxima! subgroups of ｙ ･are all normal in Ｇ.
Proof. Let Nl be an arbitrary maximaトsubgroup of Ｎ and Ａ be the normalizer of N, in G. Ｈ every maximal subgroup Gにof Ｇ containing iV, a】ways contains Ｎ，
･hen by Theorem ５，ψ肖(Ａ)＝Ｎ１φ(ｊ)ユｙ would h6ぼ Then we could have an ｅｌｅ･
. merit, n of ｙ such that 7zＥの(G) and ･,z百A', and so φ(Ｎ)百n, this conflicts with ψ(Ｇ) ｎｙ＝φ(N). Therefore Ｇ has ａ maximal subgroup G, such that G=NG,, C:石⊃Ｎ,． Since ｙ and Gi are normal in Ｇ， ｙt＝Ｇｌｎｙ is normaトin Ｇ.
Corollary ４. Let G be nilpotent and ｙ its normal subgroup. If 0(G)nA'' =
】, then every subgroup
of 7ｖis normal
in Ｇ.
Proof.
Letルf be a subgroup
of jV and ｙ≒Noﾆ)ｙl⊃ｙ2⊃
sequence
of subgroups of 7V where ｙ， is ｍａχimalin Ah-1.
Since
2) Cf. (1), p. 162
‥‥⊃Ｎ．,= M be ａ
旦旦ubgroup of finitenilpotent Groups (S. Fujita) 41 TV, is normal in Ｇ by Theorem 6. Then, sinceの(Ｇ)∩Ｎ１＝φ(ｙ1), Ｎ， Vs normal in Ｇ also. For repeating the argument with N。Ｎ。..･･。Ｎ。-1 in turn, we see yχfis nor･ mal in Ｇ.
Corollary 5. Let G be nilpotent and ｙ its normal subgroup. If there exists ａ subgroup H of Ｇ such that G=NH, NnH=l, then the maximal subgroups of ｙ are all normal in Ｇ if and only if 0(G)nN=0(.N).
Proof. ｌ￡（Ｄ(Ｇ)ｎＮ＝の(AO, then by Theorem ６ the maximal subgroups of Ｎ are all normal in Ｇ.
Conversely, if the maximal subgroups of ｙ are all normal in Ｇ， then Ｎ^Ｈ ａｎｄ ＮＨ, where Ｎｉ ａndＨ, ａre arbitrary maximal subgroups of ｙ and Ｈ ｌｅspectively, are
maximal in Ｇ. Hence we get ゆ(Ｇ)Ξ(Z)(Ｎ)Ｈａｎｄゆ(Ｇ)こＮψ(Ｈ)ａｎｄ so ゆ(Ｇ)瓜ψ
(N)0(H'). On the other hand, by Theorem ４， ψ(Ｇ)２の(jV)の(Ｈ).ヽThus we have ゆ
(Ｇ)＝の(N)の(H) and so ゆ(Ｇ)ｎＮ＝の(ｙ).
Corollary ６. Let the normal subgroup ｙ of ａ nilpotent group G be the direct product of groups of prime order which are normal in Ｇ･Then G has a subgroup 77 such that Ｇ=ＮＨ, Ｎr＼H=[ if and only if 0(.G)nN = i.
Proof. By assumption about j＼≒ φ(Ｎ)＝1 and the maximal subgroups of N are all normal in C. Hence by Corollary 5，ψ(Ｇ)ｎＮ＝1 holds.
Converse】y, since ｙ is abeliari ａｎｄの(Ｇ)ｎＮ＝1，Ｇ has ａ subgroup H such that Ｇ＝ＮＨ，Ｎｎ耳＝13).
Theorem 7. Let ｙ be a nil potent normal subgroup of Ｇ and Ｎ (Z0(G), then there exists ａ subgroup 尺such that Ｇ＝ＮＫ, Ｎ∩尺こゆ(Ｇ).
Proof. Since NΦ(G) is normal in G, 0(NΦ(G)) is normal in G. Hence
の(Ｇ/の(Ｇ))Ξ0(N0CG)/の(Ｇ))ａｎｄ so φ(ｙゆ(Ｇ)/の(Ｇ))＝ゆ(Ｇ)/の(Ｇ)･holds.
ｙの(Ｇ)/の(G) is nilpotent and so Ｎの(Ｇ)/の(Ｇ)is abelian. Moreoverの(GZΦ(Ｇ))∩ ｙの(Ｇ)Z(1)(Ｇ)＝の(Ｇ)/の(Ｇ)sｏ we. have ａ factor groupＫｉｄ)(G) such that Ｇ/の(Ｇ)＝
N0(G)/<1>(G:)･尺/の(G), Nφ(Ｇ)∩尺＝･0(G). This showｓ Ｇ = ＮＫ, Ｎｎ尺瓜ψ(Ｇ)．
田岡
References
W. Gaschijtz, Uber die (P-Untergruppe endlicher Gruppen. Math. Zeit., vol. 58, 160-170(1953) H. Zassenhaus, Lehrbuch der Gruppentheorie, vol. 1. (1937)
(Received June 25, 1960)
