On the triple product property for subgroups of finite nilpotent groups of class $2$

Abstract: A number of upper bounds are proved relating to the triple product property (TPP) for subgroups of finite nilpotent groups of class $2$. The TPP is the property defined for three non-empty subsets $S, T, U$ of a group $G$ that the group equation $s&#39;s^{{-1}t&#39;t^{{-1}u&#39;u^{-1}}} = 1$, over pairs of elements $s&#39;, s \in S$, $t&#39;, t \in T$, $u&#39;, u \in U$, is satisfied if and only if $s&#39; = s$, $t&#39; = t$, $u&#39; = u$. When $G$ is finite, and the parameter $ρ_0(G)$, called \emph{subgroup TPP ratio}, is defined as $ρ_0(G) := \max\frac{|S||T||U|}{|G|}$, where the maximum is over the collection of all triples of subgroups $S, T, U$ of $G$ satisfying the TPP, this paper proves that \textup{(1)} $ρ_0(G) &lt; \sqrt{|G:Z(G)}$} for (all) groups of nilpotency class $2$, \textup{(2)} $ρ_0(G) \leq p$ for $p$-groups with a cyclic commutator subgroup of order $p$, \textup{(3)} $ρ_0(G) = 1$ for $p$-groups of nilpotency class $2$ with a "large" centre, loosely defined as those satisfying $p² \leq |G:Z(G)| \leq p<sup>3$, \textup{(4)} and $ρ_0(G) = 1$ for $p$-groups of nilpotency class $2$ with "small" (irreducible, complex) character degrees of $1$ or $p$.

• The paper proves that every finite group of nilpotency class 2 satisfies the strict bound ρ₀(G) < √|G:Z(G)|, using induction, quotient arguments, and structural analysis of subgroup intersections with the centre.
• The paper establishes the sharp bound ρ₀(G) ≤ p for nonabelian p-groups with commutator subgroup of order p, with equality attained by an extraspecial group of order 32.
• The paper proves ρ₀(G) = 1 for class-2 p-groups with p² ≤ |G:Z(G)| ≤ p³ and for groups whose irreducible character degrees are {1,p}, showing that broad structural conditions eliminate nontrivial subgroup TPP gains.

Background and motivation

The triple product property (TPP), introduced by Cohn and Umans as a group-theoretic framework for fast matrix multiplication (2602.15796), concerns triples of non-empty subsets $S, T, U$ of a group $G$ such that the equation $s's^{-1}t't^{-1}u'u^{-1} = 1$ holds only trivially. When $S, T, U$ are subgroups, the condition simplifies to $S \cap TU = T \cap U = \{1\}$. The subgroup TPP ratio of a finite group is

$$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$

and $\rho_0(G) = 1$ for every abelian group, since the triple product map is then injective. Values of $\rho_0(G)$ exceeding $1$ correspond to non-trivial "realisations" of matrix multiplication and enter the pseudoexponent $\alpha$ of Cohn–Umans, so upper bounds on $G$0 translate into lower bounds on that parameter. The paper proves four upper bounds conjectured from computational data on small groups (computed with GAP, following Hedtke–Murthy's search algorithms): a universal bound for nilpotency class 2, a sharp bound for $G$1-groups with cyclic commutator subgroup of order $G$2, and two exact results $G$3 for classes of $G$4-groups of class 2.

The proofs rest on a common structural principle: in a non-trivial subgroup TPP triple, no member can be normal in $G$5, and no member can contain the centre $G$6 (otherwise the Neumann quotient lemma forces $G$7 for an abelian quotient $G$8). Consequently, in class-2 groups the analysis reduces to how the triple members intersect the centre.

The universal bound for class-2 groups

The main theorem states that if $G$9 has nilpotency class 2, i.e. $s's^{-1}t't^{-1}u'u^{-1} = 1$0, then

$s's^{-1}t't^{-1}u'u^{-1} = 1$1

The proof is an induction on $s's^{-1}t't^{-1}u'u^{-1} = 1$2. Given a maximal non-trivial subgroup triple $s's^{-1}t't^{-1}u'u^{-1} = 1$3, one considers $s's^{-1}t't^{-1}u'u^{-1} = 1$4. If $s's^{-1}t't^{-1}u'u^{-1} = 1$5, the quotient lemma applied to $s's^{-1}t't^{-1}u'u^{-1} = 1$6 (which remains of class 2) and the index lemma relating $s's^{-1}t't^{-1}u'u^{-1} = 1$7 to $s's^{-1}t't^{-1}u'u^{-1} = 1$8 yield the bound. If $s's^{-1}t't^{-1}u'u^{-1} = 1$9, then $S, T, U$0 is abelian and non-normal, and $S, T, U$1 is an abelian normal subgroup of $S, T, U$2; the subgroup $S, T, U$3 is normal of class 2. If $S, T, U$4, the split proposition gives $S, T, U$5. If $S, T, U$6, then either some intersection $S, T, U$7 is non-trivial and the quotient argument applies, or else $S, T, U$8 is a semidirect product forcing $S, T, U$9 and

$S \cap TU = T \cap U = \{1\}$0

The bound is consistent with the character-theoretic fact that irreducible character degrees satisfy $S \cap TU = T \cap U = \{1\}$1, and it is strict — no class-2 group in the data attains equality.

Sharp bound $S \cap TU = T \cap U = \{1\}$2 for $S \cap TU = T \cap U = \{1\}$3-groups with $S \cap TU = T \cap U = \{1\}$4

If $S \cap TU = T \cap U = \{1\}$5 is a nonabelian $S \cap TU = T \cap U = \{1\}$6-group with cyclic commutator subgroup of order $S \cap TU = T \cap U = \{1\}$7 (a class including all extraspecial groups), the class-2 bound gives only $S \cap TU = T \cap U = \{1\}$8 for $S \cap TU = T \cap U = \{1\}$9. The paper proves the much stronger, and sharp, bound $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$0.

The induction mirrors the class-2 argument, with an additional twist in the terminal case $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$1: comparing the semidirect product decompositions $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$2 and $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$3 forces $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$4. The commutator identity $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$5 then shows $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$6, so $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$7 is extraspecial, with $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$8 for $$\rho_0(G) := \frac{\beta_0(G)}{|G|}, \qquad \beta_0(G) := \max\{|S||T||U| : (S,T,U) \text{ a subgroup TPP triple of } G\},$$9. The Frattini argument then yields $\rho_0(G) = 1$0, which is impossible by the paper's lemma that two subgroups of order $\rho_0(G) = 1$1 in an extraspecial group of order $\rho_0(G) = 1$2 never generate $\rho_0(G) = 1$3 (proved via a central product decomposition into factors of order $\rho_0(G) = 1$4). Sharpness is witnessed by an extraspecial group of order 32 (GAP ID [32, 49]), the only group in the data with $\rho_0(G) = 1$5.

Two classes with $\rho_0(G) = 1$6

For $\rho_0(G) = 1$7-groups of class 2 with a "large" centre, defined by $\rho_0(G) = 1$8, the paper proves $\rho_0(G) = 1$9. The argument is short: such groups have $\rho_0(G)$0 elementary abelian of order $\rho_0(G)$1 or abelian of order $\rho_0(G)$2, and in either case $\rho_0(G)$3 contains an abelian maximal normal subgroup of index $\rho_0(G)$4, which by earlier work forces $\rho_0(G)$5. This is strictly stronger than the class-2 bound $\rho_0(G)$6 that Theorem on cyclic commutators would give.

The second exact result concerns $\rho_0(G)$7-groups of class 2 with character degree set $\rho_0(G)$8. By the Isaacs–Passman structure theorem, such a group either has an abelian maximal normal subgroup of index $\rho_0(G)$9 (giving $1$0) or has $1$1, in which case the large-centre theorem applies. Hence $1$2 in all cases. The paper notes the heuristic connection — small character degrees reflect the existence of large abelian subgroups, which in turn suppress $1$3 — but the implication runs through the classification rather than through any direct character-theoretic bound.

Limitations and open questions

The results concern only subgroup TPP triples; the more general ratio $1$4 over arbitrary subsets is not bounded here, and the conjecture $1$5 for groups with cyclic normal subgroups of prime index remains open from prior work. The class-2 bound $1$6 is not known to be tight, and the paper does not determine whether the gap between $1$7 and $1$8 can be closed for general class-2 $1$9-groups with larger commutator subgroups. All four theorems are proved for complex irreducible character degrees in the last case, and the computational evidence is limited to groups of small order (up to 128 for 2-groups, 27 for 3-groups), so the conjectures were extrapolated from sparse data. The connection to the pseudoexponent $\alpha$0 and matrix multiplication exponents is noted but not developed quantitatively.

Conclusion

The paper establishes a strict universal bound $\alpha$1 for all finite groups of nilpotency class 2, a sharp bound $\alpha$2 for $\alpha$3-groups with cyclic commutator subgroup of order $\alpha$4 (including extraspecial groups, where the bound is attained), and exact values $\alpha$5 for class-2 $\alpha$6-groups with large centre or with character degrees in $\alpha$7. Together these confirm four conjectures drawn from computational search and delineate substantial classes of groups for which subgroup TPP triples are provably trivial in size.