Focal subgroup theorem

This is a purely group-theoretic statement whose proof requires the use of some more advanced machinery, namely that from: linear representation theory

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Sylow subgroup) must also satisfy the second subgroup property (i.e., subgroup whose focal subgroup equals its intersection with the commutator subgroup)
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Statement

Let $P$ be a $p$-Sylow subgroup of a finite group $G$ and let:

$P_0=⟨x{y}^{-1}\mid x,y\in P,\mathrm{\exists }g\in G,gx{g}^{-1}=y⟩$.

In other words, $P_0$ is the focal subgroup of $P$ in $G$.

Then:

$P\cap G^{\prime }=P_0$

In other words, $P$ is a subgroup whose focal subgroup equals its intersection with the commutator subgroup.

Proof

Given: A finite group $G$, a $p$-Sylow subgroup $P$. $P_0$ is the focal subgroup of $P$, defined by:

$P_0=⟨x{y}^{-1}\mid x,y\in P,\mathrm{\exists }g\in G,gx{g}^{-1}=y⟩$.

Further, we have:

$G^{\prime }=[G,G]=⟨x{y}^{-1}\mid x,y\in G,\mathrm{\exists }g\in G,gx{g}^{-1}=y⟩$.

To prove: $P_0=P\cap G^{\prime }$.

Initial observation

First, note that $P^{\prime }\subseteq P_0\subseteq P\cap G^{\prime }$. The first inclusion is because every commutator of elements in $P$ is contained inside the generating set for $P_0$, and the second inclusion is because every element in the given generating set for $P_0$ is both in $P$ and in $G_0$.

Since $P_0$ contains $P^{\prime }$, $P/P_0$ is an Abelian group.

Main proof

Let $p_1,p_2,\dots ,p_m$ be the distinct prime divisors of $\left|G\right|$ with $p=p_1$. For each $p_i$, pick a $p_i$-Sylow subgroup $P_i$ such that $P_1=P$.

Now, any $g\in G$ can be expressed as a product $g=g_1{g}_2\dots g_m$ where each $g_i$ is conjugate to some $h_i\in P_i$ and the $g_i$ commute pairwise. If $g_1$ is also conjugate to ${h_{1^{\prime }}}^{\prime }\in P_1$ then $h_1{h}_1{\text{'}}^{-1}\in P_0$.

Let $\lambda$ be a linear character of $P$ whose kernel contains $P_0$. We claim that the function:

$\theta (g)=\lambda (h_1)$

where $h_1$ is as described above, is well-defined. The reason is that if $h_1$ and ${h_{1^{\prime }}}^{\prime }$ are two possibles, then $\lambda (h_1{h}_1{\text{'}}^{-1})=1$ and hence $\lambda (h_1)=\lambda ({h_{1^{\prime }}}^{\prime })$.

Clearly, $\theta$ is a class function on $G$. Then by the characterization of linear characters lemma, we conclude that $\theta$ is a linear character of $G$. Further $\theta$ extends $\lambda$, viz the restriction of $\theta$ to $P$ is $\lambda$.

Thus, given any linear character on $P/P_0$, we get a linear character on $G$ extending it. In particular, this means that for any element in $P\setminus P_0$, there is a linear character on $G$ that is not 1 at that element. Thus, that element cannot lie in $G^{\prime }$, and hence $P_0=P\cap G^{\prime }$.

References

Journal references

• Focal series in finite groups by Donald Gordon Higman, , Volume 5, Page 477 - 497(Year 1953): ^{More info}

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 250, Theorem 3.4, Section 7.3 (Transfer and the focal subgroup)