The Thompson subgroup \(J(P)\) of a finite \(p\)-group \(P\) is generated by the abelian subgroups of maximum order. It compresses a potentially large family of nonnormal maximal abelian subgroups into one characteristic subgroup that can be tracked in Sylow normalizers. Thompson used it to reduce normal \(p\)-complement criteria involving every nontrivial characteristic subgroup to a small number of canonical local subgroups. [@isaacs2008]

Definition

For a finite \(p\)-group \(P\), set

\[ d(P)=\max\{|A|\mid A\leq P,\ A\text{ abelian}\} \]

and

\[ \mathcal A(P)=\{A\leq P\mid A\text{ abelian and }|A|=d(P)\}. \]

Every automorphism preserves subgroup order and abelianness, so it permutes \(\mathcal A(P)\). Therefore

\[ J(P)\operatorname{char}P. \]

Fundamental properties

If \(P\) is abelian, then \(\mathcal A(P)=\{P\}\) and \(J(P)=P\). In general \(J(P)\) need not be abelian; only its generating subgroups are.

Thompson replacement

Let \(A\in\mathcal A(P)\) and suppose its position relative to another \(p\)-subgroup or a coprime acting group is inconvenient. A replacement argument finds \(A^*\in\mathcal A(P)\) with a larger normalizer, stronger centralization, or a larger prescribed intersection, without reducing its order.

Replacement is a method rather than one universal formula. It explains why \(J(P)\) makes maximum abelian subgroups accessible to local normalizers.

Thompson's normal p-complement theorem

Isaacs also proves the stronger but less economical criterion: if \(N_G(X)\) has a normal \(p\)-complement for every nontrivial characteristic subgroup \(X\operatorname{char}P\), then \(G\) does. The role of \(J(P)\) is to reduce a large family of \(X\) to the two canonical control objects \(Z(P)\) and \(J(P)\).

Frobenius kernels

Let \(N\) be a Frobenius kernel with complement \(H\). To prove that \(N\) is nilpotent, one seeks a normal \(p\)-complement in \(N\) for each odd prime \(p\). The fixed-point-free coprime action of \(H\) strongly constrains normalizers of characteristic subgroups of \(P\in\operatorname{Syl}_p(N)\). Thompson's criterion collects those constraints in \(C_N(Z(P))\) and \(N_N(J(P))\). Induction gives normal \(p\)-complements in both local subgroups and hence in \(N\). Repeating this prime by prime makes every Sylow subgroup normal, so \(N\) is nilpotent.

Examples

ZJ theory and later variants

The center

\[ Z(J(P)) \]

is often easier to normalize than \(J(P)\) itself because it is an abelian characteristic layer. Glauberman's \(ZJ\) theorem, under odd-prime, \(p\)-stability, and section-exclusion hypotheses, makes \(Z(J(P))\) normal in an appropriate quotient or supplies a normal \(p\)-complement criterion. This became a major local-analysis step in finite simple group theory.

Computation

Naively enumerating all abelian subgroups can be expensive. Practical algorithms first bound \(d(P)\) and search along a normal or central series for candidates attaining it. Replacement theorems can restrict the search to candidates normalized by selected characteristic subgroups. Maximum-elementary-abelian and maximum-order versions must be implemented separately.

This article uses the maximum-order convention of Isaacs and integrates containment stability, replacement, the normal-complement criterion, and the Frobenius-kernel application from Chapter 7. The definition and historical attribution were cross-checked against Thompson's papers, the Encyclopedia of Mathematics, and the cited public encyclopedia revision. [@isaacs2008] [@thompson1960] [@thompson1969] [@eom-thompson] [@wikipedia-thompson]