Subnormality is permuting join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) satisfying a subgroup metaproperty (i.e., permuting join-closed subgroup property)
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Statement with symbols
Suppose and are subnormal subgroups of a group , and suppose further that , i.e., the product of subgroups is again a subgroup. Then, is a subnormal subgroup of . Further, if the subnormal depth of is and the subnormal depth of is , the subnormal depth of is bounded by:
By symmetry, it is also bounded by the function . When , is the smaller of the two numbers, hence the more relevant bound.
- Subnormality is normalizing join-closed
- Normality is strongly join-closed
- 2-subnormality is conjugate-join-closed
- Join of normal and subnormal implies subnormal of same depth
- Commutator subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property
- Nilpotent commutator subgroup implies subnormal join property
|Subnormal depth of||Subnormal depth of||Upper bound on subnormal depth of given by this statement||Best known bound on subnormal depth of||Lowest possible upper bound based on explicit example|
|1||-- take trivial.|
|2||2||12||4 (see 2-subnormal implies join-transitively subnormal)||3 (see 2-subnormality is not finite-join-closed)|
|2||(see 2-subnormal implies join-transitively subnormal)||?|
- Modular property of groups: If such that we have .
- Subnormality is normalizing join-closed: If are subnormal of depths , and , then is a subnormal subgroup of depth at most .
- Subnormality satisfies intermediate subgroup condition: If and is -subnormal in , is -subnormal in .
- Subnormality is finite-relative-intersection-closed: Suppose is -subnormal in and such that is -subnormal in some subgroup of containing both and . Then, is -subnormal in .
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Given: A group , a subnormal subgroup of subnormal depth in , a subnormal subgroup of subnormal depth , such that . Let .
To prove: is a subnormal subgroup of with subnormal depth at most .
Proof: We break the proof into many steps. Note that we use permutability where we use the fact that the product is the subgroup generated by and .
The key idea is to construct one subnormal series, first going down the series, and then going back up the series using some bounds. Our first few steps are in the ambient group , and our later steps shift focus to the ambient group .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation||Commentary|
|1||is -subnormal in||Fact (3)||is -subnormal in||--||Piece together.||Shifting to ambient group .|
|2||Define a subnormal series for in by , and . Note that||Still in ambient group , viewing subnormality there.|
|3||. Moreover, each is -subnormal in .||Steps (1), (2)||This follows from the definition of subnormal depth.||Still in ambient group , viewing subnormality there.|
|4||is -subnormal in||Fact (4)||is -subnormal in||Step (3)||Piece together. In Fact (4), set .||Crucial step, where we switch to ambient group .|
|5||Fact (1)||and permute||Step (2)||[SHOW MORE]||Preparing the ground for moving up from to via s in ambient group .|
|6||normalizes||Step (2)||[SHOW MORE]||Preparing the ground for moving up from to via s in ambient group .|
|7||If has subnormal depth , then is subnormal of depth at most||Fact (2)||--||Step (4)||[SHOW MORE]||Single step up the ladder from to via s, in ambient group .|
|8||The final result||is -subnormal in )||Steps (2), (7)||[SHOW MORE]||Combine multiplicative factors accumulated in each step of the climb up.|