Subnormality is normalizing join-closed
From Groupprops
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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Statement
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 .
Related facts
- Join of normal and subnormal implies subnormal of same depth
- 2-subnormality is conjugate-join-closed
- Subnormality is permuting join-closed
Facts used
- 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.
Proof
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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 .
Proof:
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 . |
References
Textbook references
- 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}