Pronormal and subnormal implies normal
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property must also satisfy the second subgroup property
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This page describes additional conditions under which a subgroup property implication can be reversed, viz a weaker subgroup property, namely Subnormal subgroup (?), can be made to imply a stronger subgroup property, namely normal subgroupTemplate:Semitrivial result
View other subgroup property implicationreversing conditions
Contents
Statement
Verbal statement
A subgroup of a group that is both pronormal and subnormal must be normal.
Related facts
 Pronormal implies intermediately subnormaltonormal: This can further be generalized to polynormal implies intermediately subnormaltonormal.
 Normalizer of pronormal implies abnormal
 Pronormal implies selfconjugatepermutable (finite groups)
Facts used
 Pronormality satisfies intermediate subgroup condition: A pronormal subgroup in the whole group is also pronormal in any intermediate subgroup (this is used in the first proof).
Proof
The proof involves two steps.
Pronormal and 2subnormal implies normal
A 2subnormal subgroup is a subgroup realized as a normal subgroup of a normal subgroup. We prove that a pronormal 2subnormal subgroup is normal.
Given: , and is pronormal in .
To prove: is normal in . Take . We must show that .
Clearly must lie inside the normal closure of in , and since is a normal subgroup containing , must lie inside . Now, since is pronormal in , and are conjugate subgroups in , and thus are conjugate subgroups inside .
This, combined with the fact that is normal inside , tells us that .
Overall proof
The overall proof proceeds by induction, using the above lemma. The induction statement for says:
Any pronormal subgroup of subnormal depth at most is normal.
The statement is proved by showing that any pronormal subgroup of subnormal depth is of subnormal depth at most , and then appealing to induction. Here's how. Suppose is pronormal in and:
, being pronormal in is also pronormal in every intermediate subgroup (propertytheoretically, pronormality satisfies the intermediate subgroup condition). So is pronormal in . But is also 2subnormal in , and the previous lemma tells us that any pronormal 2subnormal subgroup is normal. So , and we get:
Thus has subnormal depth at most , and induction applies.
Converse
The converse is clearly true: any normal subgroup is pronormal as well as subnormal.