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Archiv der Mathematik

, Volume 80, Issue 3, pp 225–234 | Cite as

Finite groups with trivial Frattini subgroup

  • L.-C. Kappe
  • J. Kirtland
Original paper

Abstract.

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.

Mathematics Subject Classification (2000): Primary: 20E34; Secondary: 20E15. 

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Copyright information

© Birkhäuser Verlag, Basel, 2003

Authors and Affiliations

  • L.-C. Kappe
    • 1
  • J. Kirtland
    • 2
  1. 1.Department of Mathematical Sciences, State University of New York at Binghamton, Binghamton, NY 13902-6000, USAUS
  2. 2.Department of Mathematics, Marist College, Poughkeepsie, NY 12601, USAUS

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