2-subnormal not implies hypernormalized: Difference between revisions
(New page: {{subgroup property non-implication| stronger = 2-subnormal subgroup| weaker = hypernormalized}} ==Statement== ===Verbal statement=== A 2-subnormal subgroup of a group need not be [...) |
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{{subgroup property non-implication| | {{subgroup property non-implication| | ||
stronger = 2-subnormal subgroup| | stronger = 2-subnormal subgroup| | ||
weaker = hypernormalized}} | weaker = hypernormalized subgroup}} | ||
==Statement== | ==Statement== | ||
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A [[2-subnormal subgroup]] of a group need not be [[hypernormalized subgroup|hypernormalized]]. | A [[2-subnormal subgroup]] of a group need not be [[hypernormalized subgroup|hypernormalized]]. | ||
==Related facts== | |||
===Stronger facts=== | |||
* [[Abnormal normalizer and 2-subnormal not implies normal]]: In fact, the same example used here works for that as well. | |||
==Proof== | ==Proof== | ||
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===An example in the symmetric group on four letters=== | ===An example in the symmetric group on four letters=== | ||
Let <math>G</math> be the [[symmetric group:S4|symmetric group on four letters]] <math>\{ 1,2,3,4\}</math> and <math>H</math> be the two-element subgroup generated by <math>(13)(24)</math>. | Let <math>G</math> be the [[particular example::symmetric group:S4|symmetric group on four letters]] <math>\{ 1,2,3,4\}</math> and <math>H</math> be the two-element subgroup generated by <math>(13)(24)</math>. | ||
Then, <math>H</math> is normal in the subgroup <math>K = \{ (), (12)(34), (13)(24), (14)(23)\}</math>, which is normal in <math>G</math>. So <math>H</math> is 2-subnormal in <math>G</math>. | Then, <math>H</math> is normal in the subgroup <math>K = \{ (), (12)(34), (13)(24), (14)(23)\}</math>, which is normal in <math>G</math>. So <math>H</math> is 2-subnormal in <math>G</math>. | ||
On the other hand, the normalizer <math>N_G(H)</math> is a [[dihedral group:D8|dihedral subgroup of order eight]], which is a self-normalizing subgroup. | On the other hand, the normalizer <math>N_G(H)</math> is a [[dihedral group:D8|dihedral subgroup of order eight]], which is a self-normalizing subgroup. | ||
Latest revision as of 22:08, 19 September 2008
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., 2-subnormal subgroup) need not satisfy the second subgroup property (i.e., hypernormalized subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about 2-subnormal subgroup|Get more facts about hypernormalized subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property 2-subnormal subgroup but not hypernormalized subgroup|View examples of subgroups satisfying property 2-subnormal subgroup and hypernormalized subgroup
Statement
Verbal statement
A 2-subnormal subgroup of a group need not be hypernormalized.
Related facts
Stronger facts
- Abnormal normalizer and 2-subnormal not implies normal: In fact, the same example used here works for that as well.
Proof
An example in the symmetric group on four letters
Let be the symmetric group on four letters and be the two-element subgroup generated by .
Then, is normal in the subgroup , which is normal in . So is 2-subnormal in .
On the other hand, the normalizer is a dihedral subgroup of order eight, which is a self-normalizing subgroup.