Alperin's fusion theorem in terms of conjugation families

History

This statement was formulated and proved by Jonathan Lazare Alperin, in 1965.

Statement

General form

Suppose $G$ is a finite group, $S$ is a $p$-Sylow subgroup for a prime $p$ Suppose ${\mathcal {F}}$ is a collection of subgroups of $S$ with the property that for any subgroup $T$ of $S$, there exists $U\in {\mathcal {F}}$ such that:

• $U$ and $T$ are conjugate subgroups inside $G$
• $N_{S}(U)$ is a $p$-Sylow subgroup of $N_{G}(U)$

Then ${\mathcal {F}}$ is a conjugation family for $S$ in $G$.

More specific form

This form states:

Let ${\mathcal {F}}$ be the collection of all subgroups of $S$ whose normalizer in $S$ is a Sylow subgroup of the normalizer in $G$. Then, ${\mathcal {F}}$ is a conjugation family for $S$ in $G$.

This follows from the more general form, and the fact that every p-subgroup is conjugate to a p-subgroup whose normalizer in the Sylow is Sylow in its normalizer.

Proof

Key idea: induction on index

We prove the result by inducting on the index of the subgroup $\langle A\rangle$ in $S$. Suppose $T=\langle A\rangle$ and $V=\langle B\rangle$.

Base case for induction

The base case of induction is when $T=S$. By the conditions, ${\mathcal {F}}$ contains a conjugate of $S$, so $S\in {\mathcal {F}}$. Clearly, then $g\in N_{G}(S)$, so we can set $n=1$, and $g_{1}=g$.

Induction step

The key thing to remember for the induction step is that if $T$ is a proper subgroup of $S$, then $T$ is a proper subgroup of $N_{S}(T)$ (and similarly for $V$). Thus, this step reduces to three parts:

• Go from $T$ to $N_{S}(T)$
• Use the induction to argue that we can go from $N_{S}(T)$ to $N_{S}(U)$
• Go from $N_{S}(V)$ back down to $V$

References

Journal references

• Transfer and fusion in finite groups by Jonathan Lazare Alperin and Daniel Gorenstein, Journal of Algebra, ISSN 00218693, Volume 6, Page 242 - 255(Year 1967): This paper discusses the normalizers of subgroups of a Sylow subgroup in a finite group, using the ideas of a conjugation family and Alperin's fusion theorem^{Weblink (hosted on ScienceDirect)More info}

Textbook references

• Book:GL^{More info}, Page 6-7, Theorems 3.4 and 3.5