Grün's first theorem on the focal subgroup

Statement

Suppose $G$ is a finite group and $P$ is a $p$-Sylow subgroup of $G$. Let $P_0$ be the focal subgroup of $P$ in $G$. Then:

$P_0=⟨P\cap N_G(P{)}^{\prime },P\cap Q^{\prime }\mid Q\in {\mathrm{S}}_{\mathrm{y}}(G)⟩$.

In other words, $P_0$ is generated by the intersection between $P$ and the commutator subgroup of its normalizer, along with the intersection between $P$ and the commutator subgroups of all $p$-Sylow subgroups.

Facts used

1. Focal subgroup theorem
2. Alperin's fusion theorem in terms of tame intersections

Proof

Given: A finite group $G$ with $p$-Sylow subgroup $P$ having focal subgroup $P_0$.

To prove: $P_0=P_1$ where:

$P_1=⟨P\cap N_G(P{)}^{\prime },P\cap Q^{\prime }\mid Q\in {\mathrm{S}}_{\mathrm{y}}(G)⟩$.

Proof:

Proof that $P_1\le P_0$

1. $P_1\le P$: This is clear, since all the subgroups used to generate $P_1$ are contained in $P$.
2. $P_1\le [G,G]$: All the subgroups used to generate $P_1$ are contained in the commutator subgroup of some subgroup, which in turn is contained in $[G,G]$. Thus, $P_1\le [G,G]$.
3. $P_1\le P_0$: By the previous two steps, $P_1\le P\cap [G,G]$. By the focal subgroup theorem (fact (1)), $P_0=P\cap [G,G]$, so $P_1\le P_0$.

Proof that $P_0\le P_1$

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References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 252, Theorem 4.2, Chapter 7 (Fusion, transfer and p-factor groups), Section 7.4 (Theorems of Burnside, Frobenius and Grün, ^{More info}