Pi-separable and pi'-core-free implies pi-core is self-centralizing

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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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Statement

Suppose \pi is a set of primes and G is a finite group that is separable for the prime set \pi. Further, suppose the \pi'-core of G, namely O_{\pi'}(G), is trivial. Then, the \pi-core of G, namely O_\pi(G), is a self-centralizing subgroup of G:

\! C_G(O_\pi(G)) \le O_\pi(G).

Related facts

Facts with similar proofs

Applications

Facts used

  1. Pi-separability is subgroup-closed
  2. Characteristicity is centralizer-closed
  3. Normality satisfies transfer condition
  4. Characteristicity is transitive + Characteristic implies normal
  5. 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.
  6. Normality satisfies intermediate subgroup condition
  7. Cocentral implies normal
  8. 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 \pi, a \pi-separable group G such that O_{\pi'}(G) is trivial; in other words, G has no nontrivial normal \pi'-subgroup.

To prove: C_G(O_{\pi}(G)) \le O_{\pi}(G).

Proof: Let H = O_\pi(G) and C = C_G(H). Let Z = Z(H). By definition Z = C \cap H.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Z is normal in C Fact (3) [SHOW MORE]
2 Z \le O_\pi(C) Step (2) [SHOW MORE]
3 O_\pi(C) is a normal \pi-subgroup of G Facts (2), (4) [SHOW MORE]
4 O_\pi(C) \le C \cap H = Z [SHOW MORE]
5 Z = O_\pi(C): Steps (2), (4) Step-combination direct.
6 If C is strictly bigger than Z, then K = O_{\pi,\pi'}(C) is strictly bigger than Z Fact (1) G is \pi-separable. Step (5) [SHOW MORE]
7 If C is strictly bigger than Z, there exists a nontrivial complement, say M, to Z in K Facts (5), (6) Steps (1), (6) [SHOW MORE]
8 If C is strictly bigger than Z, M is nontrivial normal Hall in K Fact (7) Steps (6), (7) [SHOW MORE]
9 If C is strictly bigger than Z, M is a nontrivial normal \pi'-subgroup of G, i.e., M \le O_{\pi'}(G) Facts (2), (4), (8) Steps (6), (7), (8) [SHOW MORE]
10 If C is strictly bigger than Z, we obtain the required contradiction to the assumption that O_{\pi'}(G) is trivial. Thus, C = Z and in particular we get C \le H, as desired. G is \pi'-core-free. Steps (1), (9) Step (9) yields a nontrivial normal \pi'-subgroup of G, so O_{\pi'}(G) 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