Subnormality is normalizing join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) satisfying a subgroup metaproperty (i.e., normalizing join-closed subgroup property)
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Suppose are subnormal subgroups, with the property that : in other words, normalizes . Then the join of subgroups is also subnormal. Moreover, the subnormal depth of is bounded from above by the products of subnormal depths of and .
- Join of normal and subnormal implies subnormal of same depth
- 2-subnormality is conjugate-join-closed
- Subnormality is permuting join-closed
- Join of normal and subnormal implies subnormal of same depth: If is normal in and is -subnormal in , then is subnormal in with subnormal depth at most .
- Normality is upper join-closed: If a subgroup is normal in two intermediate subgroups, it is normal in their join.
- Subnormality satisfies intermediate subgroup condition: More specifically, if are groups such that is -subnormal in , then is also -subnormal in .
- Subnormal subgroup has a unique fastest descending subnormal series, where the series members are obtained by taking successive normal closures.
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Given: A group , subnormal subgroups such that , i.e., normalizes . has subnormal depth and has subnormal depth .
To prove: is a subnormal subgroup, with subnormal depth at most .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||Consider the descending chain defined by , and is the normal closure of in . This is the fastest descending subnormal series for , and thus, .||Fact (4)||is -subnormal in|
|2||normalizes for all . In particular, for any , .||normalizes||Step (1)||Any subgroup of defined deterministically in terms of must be invariant under any automorphism that leaves invariant.|
|3||For each , is normal in .||Fact (2)||Steps (1), (2)||By construction, is normal in , and as observed in Step (2), normalizes , so is normal in (fact (2)).|
|4||For each , is -subnormal in||Fact (3)||is -subnormal in||Given-fact combination direct|
|5||For each , is -subnormal in||Fact (1)||Steps (3), (4)||Step-fact combination direct|
|6||is -subnormal in||Steps (1), (5)|| We have a chain: |
where each member is -subnormal in its successor. This tells us that is -subnormal in .
- A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 387, Section 13.1 (Joins and intersections of subnormal subgroups)
- Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 3, Section 1.2 (First results on joins), Theorem 1.2.1, More info