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:Semi-trivial result
View other subgroup property implication-reversing conditions
- Pronormal implies intermediately subnormal-to-normal: This can further be generalized to polynormal implies intermediately subnormal-to-normal.
- Normalizer of pronormal implies abnormal
- Pronormal implies self-conjugate-permutable (finite groups)
- 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).
The proof involves two steps.
Pronormal and 2-subnormal implies normal
A 2-subnormal subgroup is a subgroup realized as a normal subgroup of a normal subgroup. We prove that a pronormal 2-subnormal 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 .
The overall proof proceeds by induction, using the above lemma. The induction statement for says:
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 (property-theoretically, pronormality satisfies the intermediate subgroup condition). So is pronormal in . But is also 2-subnormal in , and the previous lemma tells us that any pronormal 2-subnormal subgroup is normal. So , and we get:
Thus has subnormal depth at most , and induction applies.