Pi-separable and pi'-core-free implies pi-core is self-centralizing
From Groupprops
This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a group being self-centralizing. In other words, the centralizer of the subgroup in the group is contained in the subgroup
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Contents
Statement
Suppose is a set of primes and is a finite group that is separable for the prime set . Further, suppose the -core of , namely , is trivial. Then, the -core of , namely , is a self-centralizing subgroup of :
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Related facts
Facts with similar proofs
- Solvable implies Fitting subgroup is self-centralizing
- Hall and central factor implies direct factor
Applications
Facts used
- Pi-separability is subgroup-closed
- Characteristicity is centralizer-closed
- Normality satisfies transfer condition
- Characteristicity is transitive + Characteristic implies normal
- Normal Hall implies permutably complemented: Note that this only uses the case where the normal Hall subgroup is abelian, which does not require the odd-order theorem.
- Normality satisfies intermediate subgroup condition
- Cocentral implies normal
- Equivalence of definitions of normal Hall subgroup: A normal Hall subgroup is the same thing as a characteristic Hall subgroup.
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
Given: A prime set , a -separable group such that is trivial; in other words, has no nontrivial normal -subgroup.
To prove: .
Proof: Let and . Let . By definition .
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | is normal in | Fact (3) | [SHOW MORE] | ||
2 | Step (2) | [SHOW MORE] | |||
3 | is a normal -subgroup of | Facts (2), (4) | [SHOW MORE] | ||
4 | [SHOW MORE] | ||||
5 | : | Steps (2), (4) | Step-combination direct. | ||
6 | If is strictly bigger than , then is strictly bigger than | Fact (1) | is -separable. | Step (5) | [SHOW MORE] |
7 | If is strictly bigger than , there exists a nontrivial complement, say , to in | Facts (5), (6) | Steps (1), (6) | [SHOW MORE] | |
8 | If is strictly bigger than , is nontrivial normal Hall in | Fact (7) | Steps (6), (7) | [SHOW MORE] | |
9 | If is strictly bigger than , is a nontrivial normal -subgroup of , i.e., | Facts (2), (4), (8) | Steps (6), (7), (8) | [SHOW MORE] | |
10 | If is strictly bigger than , we obtain the required contradiction to the assumption that is trivial. Thus, and in particular we get , as desired. | is -core-free. | Steps (1), (9) | Step (9) yields a nontrivial normal -subgroup of , so is trivial. |
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 228, Theorem 3.2, Section 6.3 (pi-separable and pi-solvable groups), ^{More info}