On groups satisfying the converse of Lagrange's theorem
1974 ◽
Vol 75
(1)
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pp. 25-32
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Keyword(s):
In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).
Keyword(s):
2011 ◽
Vol 111
(-1)
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pp. 67-76
1969 ◽
Vol 10
(3-4)
◽
pp. 359-362
2021 ◽
Vol 58
(2)
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pp. 147-156
Keyword(s):
2012 ◽
Vol 11
(04)
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pp. 1250064
1965 ◽
Vol 9
(1)
◽
pp. 47-58
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2008 ◽
Vol 01
(03)
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pp. 369-382
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