Vipul: New page: {{subgroup metaproperty satisfaction| property = subnormal-to-normal subgroup| metaproperty = normalizer-closed subgroup property}} ==Statement== Suppose $H$ is a [[subnormal-...

2008-10-08T19:47:29Z

New page: {{subgroup metaproperty satisfaction| property = subnormal-to-normal subgroup| metaproperty = normalizer-closed subgroup property}} ==Statement== Suppose <math>H</math> is a [[subnormal-...

New page

{{subgroup metaproperty satisfaction|
property = subnormal-to-normal subgroup|
metaproperty = normalizer-closed subgroup property}}

==Statement==

Suppose <math>H</math> is a [[subnormal-to-normal subgroup]] of a group <math>G</math>: either <math>H</math> is normal in <math>G</math> or <math>H</math> is not subnormal in <math>G</math>. Then, the [[normalizer]] of <math>H</math> in <math>G</math> is also a subnormal-to-normal subgroup of <math>G</math>.

==Related facts==

* [[Intermediately subnormal-to-normal is normalizer-closed]]
* [[Intermediately normal-to-characteristic is normalizer-closed]]
* [[Automorph-conjugacy is normalizer-closed]]
* [[Intermediate automorph-conjugacy is normalizer-closed]]

==Proof==

'''Given''': A subnormal-to-normal subgroup <math>H</math> of a group <math>G</math>, with normalizer <math>N_G(H)</math>.

'''To prove''': <math>N_G(H)</math> is also a subnormal-to-normal subgroup of <math>G</math>.

'''Proof''': If <math>N_G(H)</math> is not subnormal in <math>G</math> we are done. So, we need to prove that if <math>N_G(H)</math> is a subnormal subgroup of <math>G</math>, <math>N_G(H)</math> is normal in <math>G</math>.

If <math>N_G(H)</math> is subnormal in <math>G</math>, then, since <math>H</math> is normal in <math>N_G(H)</math>, <math>H</math> is subnormal in <math>G</math>. Since <math>H</math> is subnormal-to-normal, <math>H</math> is in fact normal in <math>G</math>, so <math>N_G(H) = G</math>. Since every group is normal as a subgroup of itself, <math>N_G(H)</math> is a normal subgroup of <math>G</math>.

Subnormal-to-normal is normalizer-closed - Revision history

Vipul: New page: {{subgroup metaproperty satisfaction| property = subnormal-to-normal subgroup| metaproperty = normalizer-closed subgroup property}} ==Statement== Suppose H is a [[subnormal-...

Vipul: New page: {{subgroup metaproperty satisfaction| property = subnormal-to-normal subgroup| metaproperty = normalizer-closed subgroup property}} ==Statement== Suppose $H$ is a [[subnormal-...