Contranormality is UL-join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., contranormal subgroup) satisfying a subgroup metaproperty (i.e., UL-join-closed subgroup property)
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Statement with symbols
Suppose is a group and is a family of subgroups indexed by , and indexing set. Further, suppose each is a contranormal subgroup of . Then, the join of all the s is contranormal in the join of all the s.
- Paranormality is strongly join-closed
- Polynormality is strongly join-closed
- Paracharacteristicity is strongly join-closed
- Polycharacteristicity is strongly join-closed
Further information: Contranormal subgroup
Given a subgroup we say that is contranormal in if any normal subgroup of containing must equal the whole of . In other words, the normal closure of in is .
- Normality satisfies transfer condition: If is normal in and , then is normal in .
Given: Group , indexing set , for all , and each is contranormal in .
To prove: The join of the (which we denote as ) is contranormal in the join of the s (which we denote as ).
Proof: By the definition of contranormality, we need to show that if is a normal subgroup of the join of , and contains each , then .
So suppose is normal in with . For each , is normal in by fact (1). Also, , and so . Thus, is a normal subgroup of containing . By contranormality of in , we get , so . Since this holds for each , we get , forcing .