A note on primitive subgroups of finite solvable groups
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In [5], Johnson introduced the primitivity of subgroups andproved that a finite group G is supersolvable if every primitive subgroupof G has a prime power index in G. In that paper, he also posed an inter-esting problem: what a group looks like if all of its primitive subgroupsare maximal. In this note, we give the detail structure of such groupsin solvable case. Finally, we use the primitivity of some subgroups tocharacterize T-group and the solvable PST0-groups.
Abstract
In [5], Johnson introduced the primitivity of subgroups andproved that a finite group G is supersolvable if every primitive subgroupof G has a prime power index in G. In that paper, he also posed an inter-esting problem: what a group looks like if all of its primitive subgroupsare maximal. In this note, we give the detail structure of such groupsin solvable case. Finally, we use the primitivity of some subgroups tocharacterize T-group and the solvable PST0-groups.
참고문헌 (7)
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- W. Guo, "On primitive subgroups of finite groups", Indian J. Pure Appl. Math., 37(6), pp.369~376 (2006)
- B. Huppert, "Endliche Gruppen. I" (1967), Springer-Verlag
- D. L. Johnson, "A note on supersoluble groups", Canad. J. Math, 23, pp.562~564 (1971)
- D. J. S. Robinson, "A note on finite groups in which normality is transitive", Proc. Amer. Math. Soc, 19, pp.933~937 (1968)
- D. J. S. Robinson, "A Course in the Theory of Groups" (1996), Springer-Verlag