Pronormality is normalizing join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) satisfying a subgroup metaproperty (i.e., normalizing join-closed subgroup property)
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Contents
Statement
Suppose are pronormal subgroups such that : in other words, normalizes . Then the join of subgroups (also the same as ) is also a pronormal subgroup.
Related facts
Similar facts: statements about normalizing join-closedness
Some facts with very similar proofs:
- Weak pronormality is normalizing join-closed
- Intermediate isomorph-conjugacy is normalizing join-closed
- Intermediate automorph-conjugacy is normalizing join-closed
Other facts about normalizing joins, but with a different kind of proof:
Related facts about pronormality
- Nilpotent join of pronormal subgroups is pronormal
- Pronormality is not join-closed
- Pronormality is not intersection-closed
Proof
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
CONVENTION WARNING: This article/section uses the right-action convention. The left and right action conventions are equally powerful and statements/reasoning here can be converted to the alternative convention (the main reason being that every group is naturally isomorphic to its opposite group via the inverse map). For more on the action conventions and switching between them, refer to switching between the left and right action conventions.
Given: A group , pronormal subgroups such that .
To prove: is pronormal in . Specifically, for any , our goal is to find a such that .
Proof: The overall problem in showing that there exists such an element is to find something that simultaneously works for both and for the same element . The solution to that is to first find an element (here called ) that works for . This original element need not work for . However, since normalizes , we can now tweak this element to fix the behavior on without altering the behavior on . The tweaking happens via multiplication by a suitable element in .
The proof has the flavor of constructing an upper triangular transformation.
Step no. | Assertion/construction | Given data used | Previous steps used | Explanation | Commentary |
---|---|---|---|---|---|
1 | There exists such that . | is pronormal in , . | From definition of pronormal. | Taken care of using the element , but no control over whether goes where it should. | |
2 | Step (1) | [SHOW MORE] | Moving to a corresponding element that stabilizes/normalizes . This reference element will be the model for the tweak element that we'll construct soon. | ||
3 | normalizes , i.e., | Step (2) | [SHOW MORE] | We capitalize on the fact that normalizes in order to move inside . | |
4 | There exists such that . | is pronormal in | From definition of pronormal. | We find the tweak element that fixes . | |
5 | In particular, obtained in Step (4) is in , so . | Steps (3), (4) | Follows directly by combining steps. | We make sure that has no effect on . | |
6 | Consider . Then | Steps (1), (4), (5) | [SHOW MORE] | We combine the elements and to make sure that both and , and hence , land where needed. | |
7 | Steps (1)--(6) | [SHOW MORE] | We check that the element thus constructed is inside the required join. |
Steps (6) and (7) clinch the proof.
References
- Finite soluble groups with pronormal system normalizers by John S. Rose, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), Volume 17, Page 447 - 469(Year 1967): ^{Weblink on Oxford Journals page}^{More info}, Proposition 1.7, Page 450