Smith Groups: Algebra, Topology & Graph Theory

• Smith groups are finitely generated abelian invariants defined via the Smith normal form, capturing cokernel and determinant properties in integer matrices.
• In graph theory, Smith groups refer to the cokernel of the adjacency matrix, while critical groups from the Laplacian reveal complementary combinatorial structure.
• In topology and bordism, Smith groups arise from linked exact sequences and fixed-point representations, offering precise invariants for smooth actions and field theories.

“Smith groups” is a context-dependent term whose meaning depends on the mathematical domain in which it is used. In algebra and combinatorics, it usually denotes the cokernel encoded by the Smith normal form of an integer matrix; in graph theory, it refers either to the Smith group of the adjacency matrix or to the closely related critical group of the Laplacian; in bordism theory, it denotes the linked bordism groups appearing in the Smith long exact sequence; and in transformation-group theory, one may define a “Smith group” as a finite group for which every smooth two-fixed-point action on a sphere forces the tangent modules at the two fixed points to be isomorphic (Stanley, 2016, Debray et al., 2024, Mizerka, 2020).

1. Algebraic meaning: Smith normal form and cokernel groups

The classical algebraic setting begins with an integer matrix $M$. Its Smith normal form is a diagonal matrix

$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$

obtained from $M$ by unimodular row and column operations, with $d_i \in \mathbb{Z}_{>0}$ and $d_i \mid d_{i+1}$. Over $\mathbb{Z}$, the cokernel decomposes as

$$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$

and the torsion summand is what many combinatorial sources call the Smith group of $M$ (Stanley, 2016). The invariant factors $d_i$ are further refined by their prime-power factorizations into elementary divisors.

This viewpoint extends beyond $\mathbb{Z}$. Over an elementary divisor domain, every matrix admits a Smith normal form, and the same cokernel interpretation persists. The companion-ring framework makes this especially explicit: if $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$0 is monic, $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$1 is its companion matrix, and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$2, then the matrix ring $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$3 is canonically isomorphic to

$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$4

so $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$5 acts as multiplication by $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$6 on $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$7, and

$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$8

(Noferini et al., 2019). In this setting the “Smith group of $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$9” is literally the module-theoretic quotient associated with multiplication by $M$0.

Two structural facts are repeatedly used across the literature. First, the product $M$1 is the gcd of the $M$2 minors of $M$3 when the base ring is a UFD, giving a direct route from determinantal data to the invariant factors (Stanley, 2016). Second, the existence of Smith normal form is sensitive to the base ring: over $M$4 it can fail, as shown by the example $M$5 (Stanley, 2016). The term “Smith group” therefore presupposes not only a matrix but also a ring over which Smith normal form exists.

2. Structured matrices, companion rings, and topological applications

A major modern use of Smith groups concerns highly structured matrices $M$6, including circulant, skew-circulant, and triangular Toeplitz matrices. The core reduction theorem states that if $M$7 is monic and

$M$8

then

$M$9

where $d_i \in \mathbb{Z}_{>0}$0. Thus the invariant factors of $d_i \in \mathbb{Z}_{>0}$1 are those of $d_i \in \mathbb{Z}_{>0}$2, together with $d_i \in \mathbb{Z}_{>0}$3 extra zeros (Noferini et al., 2019). The singular part is therefore entirely controlled by the common factor $d_i \in \mathbb{Z}_{>0}$4.

The last nonzero determinantal divisor is described by a resultant: $d_i \in \mathbb{Z}_{>0}$5 When $d_i \in \mathbb{Z}_{>0}$6, this recovers $d_i \in \mathbb{Z}_{>0}$7 (Noferini et al., 2019). This reduction is not merely formal. It turns Smith group calculations into quotient-ring arithmetic and, when $d_i \in \mathbb{Z}_{>0}$8 factors into pairwise coprime factors, allows Chinese-remainder decompositions that separate the problem into smaller blocks.

The paper applies this machinery to the first homology of Brieskorn manifolds. With

$d_i \in \mathbb{Z}_{>0}$9

the relation matrix of a cyclic presentation of $d_i \mid d_{i+1}$0 is the circulant $d_i \mid d_{i+1}$1, and hence

$d_i \mid d_{i+1}$2

If $d_i \mid d_{i+1}$3, $d_i \mid d_{i+1}$4, and $d_i \mid d_{i+1}$5, then the non-unit invariant factors are $d_i \mid d_{i+1}$6 repeated $d_i \mid d_{i+1}$7 times and $d_i \mid d_{i+1}$8 repeated $d_i \mid d_{i+1}$9 times, with $\mathbb{Z}$0 zero invariant factors. Consequently,

$\mathbb{Z}$1

(Noferini et al., 2019). In this sense the Smith group is not only an algebraic invariant of a matrix but also a concrete topological invariant.

3. Graph-theoretic Smith groups and critical groups

In graph theory, the terminology bifurcates. For a finite graph $\mathbb{Z}$2 with adjacency matrix $\mathbb{Z}$3 and Laplacian $\mathbb{Z}$4, the Smith group is

$\mathbb{Z}$5

while the critical group is the torsion subgroup of $\mathbb{Z}$6, equivalently the cokernel of any reduced Laplacian. The Smith normal form of $\mathbb{Z}$7 determines $\mathbb{Z}$8; the Smith normal form of a reduced Laplacian determines $\mathbb{Z}$9 (Ducey et al., 2015, Stanley, 2016).

For the square rook’s graph $$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$0, these groups are completely explicit. The paper proves

$$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$1

and

$$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$2

For the complement $$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$3,

$$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$4

and

$$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$5

(Ducey et al., 2015). These formulas verify Rushanan’s 1986 conjectures for the adjacency Smith groups.

For Paley graphs $$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$6, the Smith group and critical group exhibit the characteristic separation between the $$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$7-part and the $$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$8-part. If $$\operatorname{coker}(M) \cong \bigoplus_{i=1}^r \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-r},$$9 and $M$0, then

$M$1

while

$M$2

where the multiplicities $M$3 are determined by carry counts in base $M$4 via Stickelberger’s theorem on Jacobi sums (Chandler et al., 2014). The same p-adic technology reappears for Peisert graphs, where the paper determines both the Smith group and the critical group prime-by-prime; for $M$5,

$M$6

and

$M$7

with the Sylow-$M$8 part controlled by p-adic valuations of Jacobi sums arranged in $M$9 and $d_i$0 blocks (Sin, 2016).

The same pattern extends to large families of strongly regular graphs. The elementary divisors of the adjacency and Laplacian matrices are computed for polar graphs, with complete $d_i$1-primary decompositions across symplectic, orthogonal, and unitary families (Pantangi et al., 2017). The Grassmann graph $d_i$2 and its complement are analyzed prime-by-prime, including the p-adic part for the complement via incidence matrices and permutation-module techniques (Ducey et al., 2017). For van Lint–Schrijver cyclotomic strongly regular graphs, the critical group splits as

$d_i$3

while the $d_i$4-primary multiplicities are again described by carry-count combinatorics (Pantangi, 2018).

A different but related direction is the Johnson association scheme. For subset-intersection matrices $d_i$5, and more generally for every integer matrix in the $d_i$6-span of the Johnson association matrices, the Smith group is reduced to the Smith groups of finitely many small matrices $d_i$7. If $d_i$8, then

$d_i$9

where $\mathbb{Z}$0 (Ducey et al., 2023). This gives critical groups for Johnson and Kneser graphs and includes adjacency, Laplacian, signless Laplacian, and Seidel matrices in a single framework.

4. Smith groups in bordism and invertible field theories

In bordism theory, “Smith groups” refers to the linked bordism groups appearing in the Smith fiber sequence. Fix a tangential structure $\mathbb{Z}$1, a virtual bundle $\mathbb{Z}$2 of rank $\mathbb{Z}$3, and a vector bundle $\mathbb{Z}$4 of rank $\mathbb{Z}$5. The Smith homomorphism is a map

$\mathbb{Z}$6

defined geometrically by taking a transverse zero locus of a section of $\mathbb{Z}$7, spectrally as a map of Thom spectra, and cohomologically as cap or cup product with the Euler class. The paper proves that these three definitions are equivalent (Debray et al., 2024).

The decisive structural statement is the cofiber sequence

$\mathbb{Z}$8

which yields the long exact sequence

$\mathbb{Z}$9

These linked bordism groups are called the Smith groups in this setting (Debray et al., 2024).

The construction is computationally potent because it changes both the dimension and the tangential structure. The paper interprets many classical sequences as Smith sequences, including the unoriented–oriented case over $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$00, the codimension-one Spin–Pin$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$01 family, the Spin–Spin$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$02 codimension-two sequence, Wood’s sequences for $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$03 and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$04, Wall’s sequence, and Becker–Gottlieb transfer maps (Debray et al., 2024). After applying Anderson duality, the same cofiber sequence becomes a long exact sequence of groups of invertible field theories: $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$05 with the maps interpreted physically as defect anomaly matching, residual anomaly, and index anomaly (Debray et al., 2024).

5. Transformation groups and the Smith property

A different use of “Smith group” arises from the Smith question in equivariant topology. If a finite group $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$06 acts smoothly on a sphere $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$07 with exactly two fixed points $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$08 and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$09, the derivatives define real $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$10-representations on $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$11 and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$12. The Smith question asks whether these tangent $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$13-modules are always isomorphic. One may define a “Smith group” as a finite group for which the answer is affirmative in every such action (Mizerka, 2020).

The paper develops representation-theoretic tools for producing negative answers. It introduces the Smith set $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$14, the primary group $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$15, the reduced primary group $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$16, and the notion of Smith matched modules. A central induction theorem states that if $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$17 is an Oliver group, $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$18, the induction homomorphism $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$19 is a monomorphism, and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$20 are non-isomorphic Smith matched $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$21-modules with induced modules satisfying the $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$22-orientability hypothesis, then there exists a smooth two-fixed-point action of $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$23 on a standard sphere whose tangent $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$24-modules at the two fixed points are $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$25 and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$26. This yields $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$27, hence a negative answer to the Smith question (Mizerka, 2020).

The paper gives explicit failures of the Smith property. For $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$28, it constructs two $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$29-dimensional real modules

$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$30

$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$31

verifies that they are Smith matched and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$32-oriented, and concludes that there is a smooth two-fixed-point action of $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$33 on $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$34 with tangent modules $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$35 and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$36 at the fixed points (Mizerka, 2020). By a direct-product amplification, $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$37 admits a Smith exotic action on $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$38 for any finite group $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$39. The paper also proves that the solvable non-nilpotent Oliver group

$\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$40

has $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$41, so it too fails the Smith property (Mizerka, 2020).

The same paper records positive and negative comparison points. Atiyah–Bott gives an affirmative answer for cyclic groups of prime order. Bredon proves that for cyclic $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$42-power groups $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$43 there exists a threshold dimension $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$44 such that for $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$45, no Smith exotic action exists on $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$46. By contrast, Cappell–Shaneson established negative answers for cyclic groups of order divisible by $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$47 in dimension at least $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$48, and $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$49 acting on $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$50 is identified as the smallest known-dimensional exotic example (Mizerka, 2020). The transformation-group sense of “Smith group” therefore concerns a rigidity property of smooth two-fixed-point actions, not a cokernel.

6. Terminological comparison and mathematical significance

The term “Smith group” is thus genuinely polysemous. In the Smith-normal-form tradition, it is a finitely generated abelian group extracted from a presentation matrix, usually a cokernel such as $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$51, $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$52, or $\operatorname{diag}(d_1,\dots,d_r,0,\dots,0),$53 (Stanley, 2016, Noferini et al., 2019). In graph theory, this algebraic meaning coexists with the critical group, which is built from the Laplacian and whose order is controlled by the Matrix–Tree Theorem (Ducey et al., 2015). In bordism theory, the term names the three linked bordism groups in a long exact sequence generated by a Smith homomorphism (Debray et al., 2024). In transformation-group theory, it labels a property of finite groups relative to the Smith question, with Oliver groups providing a broad source of counterexamples (Mizerka, 2020).

These usages are not equivalent. The Smith group of a graph is not its critical group; the Smith groups of a bordism sequence are not cokernels of integer matrices; and a “Smith group” in the sense of the Smith question need not carry any SNF interpretation. A plausible implication is that the shared terminology reflects two independent historical lineages: one from Smith normal form and abelian-group decompositions, and another from Smith theory and fixed-point phenomena. Modern literature preserves both lineages, so the meaning of “Smith groups” must always be read from context.

Across these settings, however, the term consistently marks a passage from concrete algebraic or geometric data to a rigid group-valued invariant. That passage may be effected by invariant factors, by transverse zero loci and Thom spectra, or by tangent representations at fixed points, but in each case the resulting group packages subtle structural information in a form suitable for exact classification, comparison, and computation.