On the Frattini subgroup of a finite group

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• 作者：Stefanos Aivazidis ; ^{s.aivazidis@qmul.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Adolfo Ballester-Bolinches ^{adolfo.ballester@uv.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20D25 ; 20D10
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：470
• 期：Complete
• 页码：254-262
• 全文大小：287 K

文摘

We study the class of finite groups m>G  m> satisfying mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316303155&_mathId=si1.gif&_user=111111111&_pii=S0021869316303155&_rdoc=1&_issn=00218693&md5=f8ec60eb613d2bbb60f3d3e4b94397e8" title="Click to view the MathML source">Φ(G/N)=Φ(G)N/NmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mi mathvariant="normal">Φmi><mo stretchy="false">(mo><mi>Gmi><mo stretchy="false">/mo><mi>Nmi><mo stretchy="false">)mo><mo>=mo><mi mathvariant="normal">Φmi><mo stretchy="false">(mo><mi>Gmi><mo stretchy="false">)mo><mi>Nmi><mo stretchy="false">/mo><mi>Nmi>math> for all normal subgroups m>Nm> of m>Gm>. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.