Pi-separable and pi'-core-free implies pi-core is self-centralizing
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
View other similar statements
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 :
Facts with similar proofs
- Solvable implies Fitting subgroup is self-centralizing
- Hall and central factor implies direct factor
- 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.
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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]|
|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.|