Notes on Near-Factorizations with Dihedral Quotients

1. Introduction

Let G be a finite group with identity element 1. A pair $(S,T)$ of subsets of G is called a near-factorization of G if $\left|S\right|\ge 2$, $\left|T\right|\ge 2$, and the multiplication map

$$\mu :S\times T⟶G\setminus \left\{1\right\},\mu (s,t)=st,$$

is well-defined and bijective. Equivalently, no product $st$ with $s\in S$, $t\in T$ is equal to 1, and every element of $G\setminus \left\{1\right\}$ has exactly one such representation. We write this condition as

$$G\setminus \left\{1\right\}=S\times T.$$

Subject to the same cardinality assumptions, this is equivalent to the group-ring identity

$$ST=G-1$$

in $\mathbb{Z}\left[G\right]$; in particular,

$$\left|S\right|\left|T\right|=\left|G\right|-1.$$

(1.1)

Throughout the paper we keep the standing assumption $\left|S\right|,\left|T\right|\ge 2$ in the definition of a near-factorization. Every “if and only if” criterion below includes this condition explicitly, so that the convention is never silently dropped.

Near-factorizations are closely related to factorizations of finite groups, tilings, perfect codes, Cayley graphs and algebraic combinatorics. The nonabelian case is considerably more difficult because the ambient group ring is generally noncommutative. A foundational study of near-factorizations of finite groups appears in the work of de Caen, Gregory, Hughes and Kreher [2]; further developments can be found in [1,4,5,8].

In this paper we study near-factorizations of finite groups admitting a dihedral quotient. Let p be an odd prime and write

$$D_{2p}=\langle r,s\mid r^p=s^2=1,srs=r^{-1}\rangle .$$

(1.2)

Suppose that

$$f:G⟶D_{2p}$$

is an epimorphism. The quotient images of the two factors satisfy a rigid identity in $\mathbb{Z}\left[D_{2p}\right]$. We use this identity and the irreducible representations of $D_{2p}$ to derive restrictions on the quotient multiplicities.

The main results of the paper are the following.

(i)

We establish the quotient group-ring identity

$$\widehat{f}\left(S\right)\widehat{f}\left(T\right)=|Ker\left(f\right)|\mathbf{H}-1.$$

(ii)

We derive a linear spectral obstruction: the two quotient coset counts of each factor differ by exactly one, and the two factors have opposite imbalances. (The oddness of $\left|S\right|$ and $\left|T\right|$ already follows from $\left|S\right|\left|T\right|=\left|G\right|-1$.)

(iii)

We derive determinant-unit conditions in cyclotomic integer rings from the two-dimensional irreducible representations of $D_{2p}$. The conditions for the various representations are Galois conjugate and hence equivalent.

(iv)

We prove a Fourier energy identity and a joint energy inequality for the quotient images.

(v)

We formulate an exact finite system of nonnegative integer equations for the quotient multiplicities.

(vi)

Under an explicit orbit-commutativity hypothesis, we obtain closed formulas for the orbit norms of the quotient images.

(vii)

We prove a lifting criterion for direct products $D_{2p}\times C_m$ and give an exact verification procedure.

It is useful to record the logical hierarchy of these results. The quotient identity is an unconditional consequence of the near-factorization property, and the exact quotient coefficient system is equivalent to it at the level of coefficients. The linear, matrix and determinant-unit conditions, as well as the energy identities, are consequences of the quotient identity; they may be used as preliminary filters in computations, but they are not independent of the full coefficient system. By contrast, the orbit-norm formulas require an explicit commutativity hypothesis, which we state precisely and do not claim to be necessary.

2. Preliminaries

2.1. Group-Ring Notation

For a finite group G and a subset $A\subseteq G$, we identify A with the group-ring element

$$A=\sum _{a\in A}a\in \mathbb{Z}\left[G\right].$$

If

$$A=\sum _{g\in G}{a}_{g}g,B=\sum _{g\in G}{b}_{g}g,$$

then the coefficient of g in $AB$ is

$${\left[AB\right]}_g=\sum _{xy=g}{a}_x{b}_y.$$

Let $f:G\to H$ be a group homomorphism. Its linear extension

$$\widehat{f}:\mathbb{Z}\left[G\right]⟶\mathbb{Z}\left[H\right]$$

is a ring homomorphism. If $A\subseteq G$, then

$$\widehat{f}\left(A\right)=\sum _{h\in H}|A\cap f^{-1}\left(h\right)|h.$$

We use standard facts from the representation theory of finite groups; see, for instance, Serre [7].

Definition 1. 

Let $f:G\to H$ be a group homomorphism and let $A\subseteq G$. For $h\in H$, the number

$${\mu }_A\left(h\right)=|A\cap f^{-1}\left(h\right)|$$

is called the quotient multiplicity of A at h.

2.2. The Dihedral Group

Throughout the paper,

$$D_{2p}=\langle r,s\mid r^p=s^2=1,srs=r^{-1}\rangle .$$

Every element of $D_{2p}$ has a unique expression

$$r^a{s}^{\epsilon },a\in {\mathbb{F}}_p,\epsilon \in \{0,1\}.$$

The multiplication rule is

$$\left(r^a{s}^{\epsilon }\right)\left(r^b{s}^{\delta }\right)=r^{a+{(-1)}^{\epsilon }b}{s}^{\epsilon +\delta },$$

(2.1)

where the second exponent is reduced modulo 2.

Let

$$\mathbf{H}=\sum _{h\in D_{2p}}h$$

(2.2)

be the full group sum. Since $|D_{2p}|=2p$,

$${\mathbf{H}}^2=2p\mathbf{H}.$$

(2.3)

The rotation subgroup is

$$R=\langle r\rangle ,$$

and its other coset is

$$Rs=\{r^{a}s:a\in {\mathbb{F}}_p\}.$$

2.3. Irreducible representations

Let

$$\zeta =e^{2\pi i/p}.$$

The two one-dimensional representations of $D_{2p}$ are

$${\chi }_{+}\left(r\right)=1,{\chi }_{+}\left(s\right)=1,$$

and

$${\chi }_{-}\left(r\right)=1,{\chi }_{-}\left(s\right)=-1.$$

For $1\le j\le (p-1)/2$, define

$${\rho }_j\left(r\right)=\left(\begin{array}{cc}{\zeta }^j& 0\\ 0& {\zeta }^{-j}\end{array}\right),{\rho }_j\left(s\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$$

(2.4)

Then

$${\rho }_j\left(r^a{s}^{\epsilon }\right)=\left\{\begin{array}{cc}\left(\begin{array}{cc}{\zeta }^{ja}& 0\\ 0& {\zeta }^{-ja}\end{array}\right),\hfill & \epsilon =0,\hfill \\ \left(\begin{array}{cc}0& {\zeta }^{ja}\\ {\zeta }^{-ja}& 0\end{array}\right),\hfill & \epsilon =1.\hfill \end{array}\right.$$

(2.5)

The representations

$${\chi }_{+},{\chi }_{-},{\rho }_1,\dots ,{\rho }_{(p-1)/2}$$

form a complete set of irreducible complex representations of $D_{2p}$.

For every nontrivial irreducible representation $\theta$ of $D_{2p}$,

$$\theta \left(\mathbf{H}\right)=0.$$

(2.6)

3. The Quotient Group-Ring Identity

We begin with the basic identity from which all subsequent restrictions follow.

Theorem 1 

(Quotient identity). Let G be a finite group and let

$$G\setminus \left\{1\right\}=S\times T$$

be a near-factorization. Suppose that

$$f:G⟶D_{2p}$$

is an epimorphism. Put

$$K=Ker\left(f\right),k=\left|K\right|.$$

Then

$$\widehat{f}\left(S\right)\widehat{f}\left(T\right)=k\mathbf{H}-1\mathrm{in}\mathbb{Z}\left[D_{2p}\right].$$

(3.1)

Proof  of Theorem 2. 

The near-factorization condition gives

$$ST=G-1$$

in $\mathbb{Z}\left[G\right]$. Since $\widehat{f}$ is a ring homomorphism,

$$\widehat{f}\left(S\right)\widehat{f}\left(T\right)=\widehat{f}\left(G\right)-1.$$

Every element of $D_{2p}$ has exactly k preimages under f. Hence

$$\widehat{f}\left(G\right)=k\sum _{h\in D_{2p}}h=k\mathbf{H}.$$

Therefore

$$\widehat{f}\left(S\right)\widehat{f}\left(T\right)=k\mathbf{H}-1.$$

   □

Lemma 1. 

The element $k\mathbf{H}-1$ is invertible in $\mathbb{Q}\left[D_{2p}\right]$, with

$${(k\mathbf{H}-1)}^{-1}=-1+{\displaystyle \frac{k}{2pk-1}}\mathbf{H}.$$

(3.2)

Proof of Lemma 3. 

Using ${\mathbf{H}}^2=2p\mathbf{H}$, we have

$$\begin{array}{cc}\hfill (k\mathbf{H}-1)\left(-1+{\displaystyle \frac{k}{2pk-1}}\mathbf{H}\right)& =1-k\mathbf{H}-{\displaystyle \frac{k}{2pk-1}}\mathbf{H}+{\displaystyle \frac{2p{k}^2}{2pk-1}}\mathbf{H}\hfill \\ \hfill & =1.\hfill \end{array}$$

The same calculation in the reverse order gives the result.    □

Corollary 1. 

With the notation of Theorem 1, put

$$X=\widehat{f}\left(S\right),Y=\widehat{f}\left(T\right).$$

Then X and Y are invertible in $\mathbb{Q}\left[D_{2p}\right]$ and

$$XY=YX=k\mathbf{H}-1.$$

(3.3)

Proof of Corollary 4. 

By Theorem 1 and Lemma 1, $XY$ is invertible in $\mathbb{Q}\left[D_{2p}\right]$. The algebra $\mathbb{Q}\left[D_{2p}\right]$ is finite-dimensional over $\mathbb{Q}$; since the composition of the left-multiplication maps

$$L_X{L}_Y=L_{XY}$$

is invertible, both $L_X$ and $L_Y$ are invertible. In particular, there exists U with $XU=1$. Since $L_X$ is injective and

$$X(UX-1)=0,$$

we also have $UX=1$. Thus X is a unit, and the same argument applies to Y. Because $k\mathbf{H}-1$ is central,

$$Y=X^{-1}(k\mathbf{H}-1)=(k\mathbf{H}-1)X^{-1},$$

and consequently $XY=YX$.    □

Remark 1. 

The commutativity in Corollary 1 concerns the two specific quotient images X and Y. It does not imply that arbitrary elements of $\mathbb{Z}\left[D_{2p}\right]$ commute, nor does it imply that all automorphic translates of X and Y commute.

4. Linear Spectral Obstructions

Let

$$X=\widehat{f}\left(S\right),Y=\widehat{f}\left(T\right).$$

For $A\in \{S,T\}$ define

$$R_A=\left|\{a\in A:f\left(a\right)\in R\}\right|,$$

and

$$F_A=\left|\{a\in A:f\left(a\right)\in Rs\}\right|.$$

Thus

$$R_A+F_A=\left|A\right|.$$

(4.1)

Define the coset imbalance

$$\epsilon \left(A\right)=R_A-F_A.$$

(4.2)

Theorem 2 

(Linear spectral obstruction). Let $G\setminus \left\{1\right\}=S\times T$ be a near-factorization and let $f:G\to D_{2p}$ be an epimorphic homomorphism. Then

$$\epsilon \left(S\right)\epsilon \left(T\right)=-1.$$

(4.3)

Consequently,

$$\epsilon \left(S\right),\epsilon \left(T\right)\in \{-1,1\}.$$

(4.4)

Proof of Theorem 6. 

The character ${\chi }_{-}$ satisfies

$${\chi }_{-}\left(r^a{s}^{\epsilon }\right)={(-1)}^{\epsilon }.$$

Therefore

$${\chi }_{-}\left(X\right)=R_S-F_S=\epsilon \left(S\right),$$

and similarly

$${\chi }_{-}\left(Y\right)=R_T-F_T=\epsilon \left(T\right).$$

Since ${\chi }_{-}\left(\mathbf{H}\right)=0$, applying ${\chi }_{-}$ to (3.1) gives

$$\epsilon \left(S\right)\epsilon \left(T\right)=-1.$$

Both quantities are integers, so they must be 1 and $-1$.    □

Corollary 2. 

Under the assumptions of Theorem 2,

$$\{R_S,F_S\}=\left\{{\displaystyle \frac{\left|S\right|+1}{2}},{\displaystyle \frac{\left|S\right|-1}{2}}\right\},$$

(4.5)

and

$$\{R_T,F_T\}=\left\{{\displaystyle \frac{\left|T\right|+1}{2}},{\displaystyle \frac{\left|T\right|-1}{2}}\right\}.$$

(4.6)

Moreover, the larger of $R_S,F_S$ occurs in the opposite quotient coset from the larger of $R_T,F_T$.

Proof of Corollary 7. 

Equation (4.5) follows from

$$R_S+F_S=\left|S\right|,R_S-F_S=\pm 1.$$

The formula for T is identical. The final assertion follows from

$$\epsilon \left(S\right)\epsilon \left(T\right)=-1.$$

   □

Remark 2. 

The oddness of $\left|S\right|$ and $\left|T\right|$ already follows from the cardinality equation

$$\left|S\right|\left|T\right|=2pk-1,$$

without any representation theory. The additional content of the linear character argument is the exact coset imbalance

$$\epsilon \left(S\right),\epsilon \left(T\right)\in \{-1,1\}$$

and the fact that the two factors have opposite imbalances.

Remark 3. 

The coset-imbalance conclusion does not depend on the full dihedral structure: if a finite group G admits an epimorphism onto $C_2$ and $G\setminus \left\{1\right\}=S\times T$, then the same sign-character argument applied to the two preimage classes yields $\epsilon \left(S\right)\epsilon \left(T\right)=-1$. The dihedral quotient is used below only to combine this obstruction with the other spectral conditions.

When G is itself the dihedral group $D_{2p}$ (that is, f is the identity map), the framework of this paper yields necessary conditions for near-factorizations of $D_{2p}$, a case studied in detail by Kreher, Paterson and Stinson [3]; the present paper treats the quotient-multiplicity problem for arbitrary epimorphisms $f:G\to D_{2p}$.

5. Two-Dimensional Representations and Units in Cyclotomic Integer Rings

We next derive arithmetic restrictions from the two-dimensional representations of $D_{2p}$.

Theorem 3 

(Determinant-unit obstruction). Let $X=\widehat{f}\left(S\right)$ and $Y=\widehat{f}\left(T\right)$. For $1\le j\le (p-1)/2$, put

$$X_j={\rho }_j\left(X\right),Y_j={\rho }_j\left(Y\right).$$

Then

$$X_j{Y}_j=-I_2,$$

(5.1)

and

$$\det \left(X_j\right)\det \left(Y_j\right)=1.$$

(5.2)

In particular, $\det \left(X_j\right)$ and $\det \left(Y_j\right)$ are units in $\mathbb{Z}\left[\zeta \right]$.

Proof of Theorem 10. 

Since ${\rho }_j$ is nontrivial,

$${\rho }_j\left(\mathbf{H}\right)=0.$$

Applying ${\rho }_j$ to the quotient identity (3.1) gives

$$X_j{Y}_j=-I_2.$$

Taking determinants yields

$$\det \left(X_j\right)\det \left(Y_j\right)=1.$$

The entries of $X_j$ and $Y_j$ belong to $\mathbb{Z}\left[\zeta \right]$. Therefore their determinants are algebraic integers. Since their product is 1, each is a unit in $\mathbb{Z}\left[\zeta \right]$.    □

For $a\in {\mathbb{F}}_p$, write

$$x_{a,0}=|\{u\in S:f\left(u\right)=r^a\}|,x_{a,1}=\left|\{u\in S:f\left(u\right)=r^{a}s\}\right|.$$

Then

$$X_j=\left(\begin{array}{cc}{\displaystyle \sum _{a\in {\mathbb{F}}_p}{x}_{a,0}{\zeta }^{ja}}& {\displaystyle \sum _{a\in {\mathbb{F}}_p}{x}_{a,1}{\zeta }^{ja}}\\ {\displaystyle \sum _{a\in {\mathbb{F}}_p}{x}_{a,1}{\zeta }^{-ja}}& {\displaystyle \sum _{a\in {\mathbb{F}}_p}{x}_{a,0}{\zeta }^{-ja}}\end{array}\right).$$

(5.3)

Consequently,

$$\begin{array}{cc}\hfill \det \left(X_j\right)=& \left(\sum _{a\in {\mathbb{F}}_p}{x}_{a,0}{\zeta }^{ja}\right)\left(\sum _{a\in {\mathbb{F}}_p}{x}_{a,0}{\zeta }^{-ja}\right)\hfill \\ \hfill & -\left(\sum _{a\in {\mathbb{F}}_p}{x}_{a,1}{\zeta }^{ja}\right)\left(\sum _{a\in {\mathbb{F}}_p}{x}_{a,1}{\zeta }^{-ja}\right).\hfill \end{array}$$

(5.4)

Corollary 3. 

A necessary condition for the existence of a near-factorization with dihedral quotient $D_{2p}$ is that, for every $1\le j\le (p-1)/2$, the algebraic integer in (5.4) is a unit in $\mathbb{Z}\left[\zeta \right]$.

Remark 4. 

For $1\le j\le (p-1)/2$, let ${\sigma }_j$ be the automorphism of $\mathbb{Q}\left(\zeta \right)$ with ${\sigma }_j\left(\zeta \right)={\zeta }^j$. Then

$$\det \left({\rho }_j\left(X\right)\right)={\sigma }_j\left(\det \left({\rho }_1\left(X\right)\right)\right).$$

Consequently, the determinant-unit conditions for the various j are Galois conjugate, and hence they are equivalent: it suffices to test a single determinant exactly.

Remark 5. 

Let $K^{+}=\mathbb{Q}(\zeta +{\zeta }^{-1})$. Each

$${\delta }_j:=\det \left({\rho }_j\left(X\right)\right)={\left|\sum _{a\in {\mathbb{F}}_p}{x}_{a,0}{\zeta }^{ja}\right|}^2-{\left|\sum _{a\in {\mathbb{F}}_p}{x}_{a,1}{\zeta }^{ja}\right|}^2$$

is a real algebraic integer, and so is its inverse $\det \left({\rho }_j\left(Y\right)\right)$. Hence

$${\delta }_j\in {\mathcal{O}}_{K^{+}}^{\times }.$$

We do not claim that ${\delta }_j$ is totally positive: the displayed difference of two squared moduli may be negative. For background on units in integral group rings and related unit groups, see Sehgal [6].

Remark 6. 

The determinant conditions distinguish quotient profiles that have the same cardinalities and the same coset imbalance. They therefore provide genuinely finer information than the linear character obstruction.

6. Fourier Energy

For

$$Z=\sum _{h\in D_{2p}}{z}_{h}h\in \mathbb{C}\left[D_{2p}\right],$$

define its coefficient energy by

$$\mathcal{E}\left(Z\right)=\sum _{h\in D_{2p}}{\left|z_h\right|}^2.$$

(6.1)

For a complex matrix M, let

$${\parallel M\parallel }_{\mathrm{F}}^2=\sum _{i,j}{\left|M_{ij}\right|}^2$$

denote the squared Frobenius norm.

Theorem 4 

(Fourier energy identity). Let $X=\widehat{f}\left(S\right)$. Then

$$2p\mathcal{E}\left(X\right)={\left|S\right|}^2+{\left|\epsilon \left(S\right)\right|}^2+2{\displaystyle \sum _{j=1}^{(p-1)/2}}{\parallel {\rho }_j\left(X\right)\parallel }_{\mathrm{F}}^2.$$

(6.2)

For a near-factorization,

$$2p\mathcal{E}\left(X\right)={\left|S\right|}^2+1+2{\displaystyle \sum _{j=1}^{(p-1)/2}}{\parallel {\rho }_j\left(X\right)\parallel }_{\mathrm{F}}^2.$$

(6.3)

The analogous identities hold for $Y=\widehat{f}\left(T\right)$.

Proof of Theorem 15. 

The Plancherel identity for the finite group $D_{2p}$ gives

$$2p\mathcal{E}\left(X\right)=|{\chi }_{+}{\left(X\right)|}^2+|{\chi }_{-}{\left(X\right)|}^2+{\displaystyle \sum _{j=1}^{(p-1)/2}}2{\parallel {\rho }_j\left(X\right)\parallel }_{\mathrm{F}}^2.$$

Now

$${\chi }_{+}\left(X\right)=\left|S\right|$$

and

$${\chi }_{-}\left(X\right)=\epsilon \left(S\right).$$

This proves (6.2). Equation (6.3) follows from

$$\epsilon {\left(S\right)}^2=1$$

by Theorem 2.    □

Corollary 4 

(Energy lower bound). Under the assumptions of Theorem 4,

$$2p\mathcal{E}\left(X\right)\ge {\left|S\right|}^2+1,$$

(6.4)

and

$$2p\mathcal{E}\left(Y\right)\ge {\left|T\right|}^2+1.$$

(6.5)

Remark 7. 

The energy identity (6.2) is a consistency identity for quotient profiles: it holds for the coefficients of any element of $\mathbb{Z}\left[D_{2p}\right]$. The lower bound (6.4) is already implied by the coset-imbalance condition and the Cauchy–Schwarz inequality, and therefore does not provide an additional filter once the linear obstruction has been imposed. Indeed, putting

$$R=\sum _{a\in {\mathbb{F}}_p}{x}_{a,0},F=\sum _{a\in {\mathbb{F}}_p}{x}_{a,1},n=R+F=\left|S\right|,$$

Cauchy–Schwarz gives

$$p\sum _{a\in {\mathbb{F}}_p}{x}_{a,0}^2\ge R^2,p\sum _{a\in {\mathbb{F}}_p}{x}_{a,1}^2\ge F^2,$$

and hence

$$2p\mathcal{E}\left(X\right)\ge 2(R^2+F^2)={(R+F)}^2+{(R-F)}^2=n^2+1,$$

where the last equality uses $R-F=\pm 1$. Thus any profile that passes the linear obstruction automatically satisfies (6.4).

The following proposition records a genuinely joint condition, which uses the relation between the two factors rather than either factor alone.

Proposition 1 

(Joint energy inequality). Let $X=\widehat{f}\left(S\right)$ and $Y=\widehat{f}\left(T\right)$ arise from a near-factorization, and set

$$A=2p\mathcal{E}\left(X\right)-{\left|S\right|}^2-1,B=2p\mathcal{E}\left(Y\right)-{\left|T\right|}^2-1.$$

Then

$$AB\ge 4{(p-1)}^2.$$

(6.6)

Proof of Proposition 18. 

For $1\le j\le (p-1)/2$ put

$$a_j=\parallel {\rho }_j{\left(X\right)\parallel }_{\mathrm{F}}^2,b_j={\parallel {\rho }_j\left(Y\right)\parallel }_{\mathrm{F}}^2.$$

By (5.1),

$$2=|tr\left({\rho }_j\left(X\right){\rho }_j\left(Y\right)\right)|\le \parallel {\rho }_j{\left(X\right)\parallel }_{\mathrm{F}}{\parallel {\rho }_j\left(Y\right)\parallel }_{\mathrm{F}},$$

so that

$$a_j{b}_j\ge 4.$$

The energy identity gives

$$A=2{\displaystyle \sum _{j=1}^{(p-1)/2}}{a}_j,B=2{\displaystyle \sum _{j=1}^{(p-1)/2}}{b}_j.$$

Using the Cauchy–Schwarz inequality once more,

$$AB=4\left(\sum _j{a}_j\right)\left(\sum _j{b}_j\right)\ge 4{\left(\sum _j\sqrt{a_j{b}_j}\right)}^2\ge 4{(p-1)}^2,$$

since $\sqrt{a_j{b}_j}\ge 2$ for each of the $(p-1)/2$ values of j.    □

Remark 8. 

The joint inequality exploits the relation ${\rho }_j\left(X\right){\rho }_j\left(Y\right)=-I_2$ between the two factors, so it contains more information than the individual lower bound (6.4). It is nevertheless a consequence of the quotient identity via (5.1) and the energy identity, rather than an independent condition on quotient profiles.

7. Exact Quotient Coefficient Equations

For $a,b\in {\mathbb{F}}_p$ and $\epsilon ,\delta \in \{0,1\}$, define

$$x_{a,\epsilon }=\left|\{u\in S:f\left(u\right)=r^a{s}^{\epsilon }\}\right|,$$

(7.1)

and

$$y_{b,\delta }=\left|\{v\in T:f\left(v\right)=r^b{s}^{\delta }\}\right|.$$

(7.2)

Theorem 5 

(Exact quotient coefficient system). The quotient multiplicities of S and T satisfy

$$\sum _{\begin{array}{c}a+{(-1)}^{\epsilon }b=c\\ \epsilon +\delta \equiv \eta (mod2)\end{array}}{x}_{a,\epsilon }{y}_{b,\delta }=k-11_{(c,\eta )=(0,0)}$$

(7.3)

for every $c\in {\mathbb{F}}_p$ and every $\eta \in \{0,1\}$.

Proof of Theorem 20. 

The coefficient of $r^c{s}^{\eta }$ in

$$XY=\widehat{f}\left(S\right)\widehat{f}\left(T\right)$$

is

$$\sum _{\begin{array}{c}a+{(-1)}^{\epsilon }b=c\\ \epsilon +\delta \equiv \eta (mod2)\end{array}}{x}_{a,\epsilon }{y}_{b,\delta }$$

by the multiplication rule (2.1).

The coefficient of $r^c{s}^{\eta }$ in $k\mathbf{H}-1$ is

$$k-11_{(c,\eta )=(0,0)}.$$

The result follows.    □

Corollary 5. 

The quotient multiplicities satisfy

$$\sum _{a\in {\mathbb{F}}_p}{\displaystyle \sum _{\epsilon =0}^1}{x}_{a,\epsilon }=\left|S\right|,$$

(7.4)

$$\sum _{b\in {\mathbb{F}}_p}{\displaystyle \sum _{\delta =0}^1}{y}_{b,\delta }=\left|T\right|,$$

(7.5)

and

$$\left|S\right|\left|T\right|=2pk-1.$$

(7.6)

Proof of Corollary 21. 

Equations (7.4) and (7.5) follow from the definitions. Summing (7.3) over all c and $\eta$ gives

$$\left|S\right|\left|T\right|=2pk-1.$$

   □

Definition 2. 

Fix positive integers $p,k,n_S,n_T$, where p is an odd prime, $n_S,n_T\ge 2$, and $n_S{n}_T=2pk-1$. A pair of nonnegative integer arrays

$${\left(x_{a,\epsilon }\right)}_{a\in {\mathbb{F}}_p,\epsilon \in \{0,1\}},{\left(y_{b,\delta }\right)}_{b\in {\mathbb{F}}_p,\delta \in \{0,1\}}$$

is called a quotient-feasible profile if

$$\sum _{a,\epsilon }{x}_{a,\epsilon }=n_S,\sum _{b,\delta }{y}_{b,\delta }=n_T,$$

and the arrays satisfy the coefficient equations (7.3).

The linear condition in Theorem 2 and the determinant-unit conditions in Theorem 3 are consequences of quotient feasibility. They may nevertheless be used as preliminary filters before solving the full coefficient system.

Lemma 2. 

For every quotient-feasible profile,

$$0\le x_{a,\epsilon }\le k,0\le y_{b,\delta }\le k$$

(7.7)

for all $a,b\in {\mathbb{F}}_p$ and $\epsilon ,\delta \in \{0,1\}$.

Proof of Lemma 23. 

Since $Y\ne 0$, there exists $v\in D_{2p}$ with $y_v\ge 1$. For any $h\in D_{2p}$, the coefficient ${\left[XY\right]}_{hv}$ of $hv$ in $XY$ satisfies

$$x_h{y}_v\le {\left[XY\right]}_{hv}.$$

By the quotient identity, ${\left[XY\right]}_{hv}\le k$ for every element of $D_{2p}$. Hence $x_h\le k$; the bound for the $y_{b,\delta }$ is identical.    □

Proposition 2 

(Finite feasibility). Fix p, k, $n_S$ and $n_T$. There are only finitely many quotient-feasible profiles. The existence of a near-factorization

$$G\setminus \left\{1\right\}=S\times T$$

with quotient $D_{2p}$ implies the existence of such a profile.

Proof of Proposition 24. 

By Lemma 2, each variable $x_{a,\epsilon }$ and $y_{b,\delta }$ is a nonnegative integer bounded by k. Thus only finitely many arrays are possible. The preceding theorems give necessary equations for every near-factorization.    □

Remark 9. 

The quotient-feasibility conditions are necessary but are not sufficient in general. They record how many elements of S and T lie in each quotient fiber, but they do not record the positions of those elements inside the fibers of f.

8. Orbit Norms Under Explicit Commutativity

The previous sections do not require any commutativity assumption beyond the consequences of the quotient identity. We now describe a conditional orbit-norm reduction.

Let q be a primitive root modulo p. Define

$$\alpha \left(r\right)=r^q,\alpha \left(s\right)=s.$$

(8.1)

This extends to an automorphism of $D_{2p}$ of order $p-1$.

Definition 3 

(Orbit-commutativity). Let $X,Y\in \mathbb{Z}\left[D_{2p}\right]$. We say that $(X,Y)$ satisfies the orbit-commutativity condition if

$$[{\alpha }^i\left(X\right),{\alpha }^j\left(X\right)]=0,$$

(8.2)

$$[{\alpha }^i\left(Y\right),{\alpha }^j\left(Y\right)]=0,$$

(8.3)

and

$$[{\alpha }^i\left(X\right),{\alpha }^j\left(Y\right)]=0$$

(8.4)

for all $0\le i,j\le p-2$.

Remark 10. 

The hypothesis is partly redundant in the present setting. Put $C=k\mathbf{H}-1$, $X_i={\alpha }^i\left(X\right)$ and $Y_i={\alpha }^i\left(Y\right)$. By the quotient identity, $Y_i=X_i^{-1}C$; since C is central and invertible in $\mathbb{Q}\left[D_{2p}\right]$, the conditions $[X_i,X_j]=0$ for all $i,j$ imply $[X_i,Y_j]=0$ and $[Y_i,Y_j]=0$ automatically. We keep the symmetric formulation for clarity.

Under this hypothesis, define

$$N_{\alpha }\left(X\right)={\displaystyle \prod _{i=0}^{p-2}}{\alpha }^i\left(X\right),N_{\alpha }\left(Y\right)={\displaystyle \prod _{i=0}^{p-2}}{\alpha }^i\left(Y\right).$$

(8.5)

Lemma 3 

(Orbit invariance). If $(X,Y)$ satisfies the orbit-commutativity condition, then

$$\alpha \left(N_{\alpha }\left(X\right)\right)=N_{\alpha }\left(X\right),\alpha \left(N_{\alpha }\left(Y\right)\right)=N_{\alpha }\left(Y\right).$$

Proof of Lemma 28. 

Since ${\alpha }^{p-1}=\mathrm{id}$, applying $\alpha$ to $N_{\alpha }\left(X\right)$ cyclically permutes its factors:

$$\alpha \left(N_{\alpha }\left(X\right)\right)=\alpha \left(X\right){\alpha }^2\left(X\right)\dots {\alpha }^{p-2}\left(X\right)X.$$

The factors commute by hypothesis, so the product is unchanged: $\alpha \left(N_{\alpha }\left(X\right)\right)=N_{\alpha }\left(X\right)$. The argument for $N_{\alpha }\left(Y\right)$ is the same.    □

Theorem 6 

(Conditional orbit-norm product). Let

$$X=\widehat{f}\left(S\right),Y=\widehat{f}\left(T\right),$$

and suppose that $(X,Y)$ satisfies the orbit-commutativity condition. Then

$$N_{\alpha }\left(X\right)N_{\alpha }\left(Y\right)=1+d\mathbf{H},$$

(8.6)

where

$$d={\displaystyle \frac{{(2pk-1)}^{p-1}-1}{2p}}.$$

(8.7)

In particular d is a nonnegative integer: since $\gcd (2pk-1,2p)=1$ and $\phi \left(2p\right)=p-1$, Euler’s theorem gives ${(2pk-1)}^{p-1}\equiv 1(mod2p)$.

Proof of Theorem 29. 

Since $\alpha \left(\mathbf{H}\right)=\mathbf{H}$, the quotient identity gives

$${\alpha }^i\left(X\right){\alpha }^i\left(Y\right)=k\mathbf{H}-1$$

for every i. By the orbit-commutativity condition, all factors can be reordered, and therefore

$$N_{\alpha }\left(X\right)N_{\alpha }\left(Y\right)={\displaystyle \prod _{i=0}^{p-2}}{\alpha }^i\left(XY\right)={(k\mathbf{H}-1)}^{p-1}.$$

Using ${\mathbf{H}}^2=2p\mathbf{H}$, one obtains

$${(k\mathbf{H}-1)}^{p-1}=1+{\displaystyle \frac{{(2pk-1)}^{p-1}-1}{2p}}\mathbf{H}.$$

This proves (8.6).    □

Theorem 7 

(Explicit orbit norms). Let $X=\widehat{f}\left(S\right)$ and $Y=\widehat{f}\left(T\right)$, and suppose that $(X,Y)$ satisfies the orbit-commutativity condition. Then

$$N_{\alpha }\left(X\right)=1+{\displaystyle \frac{{\left|S\right|}^{p-1}-1}{2p}}\mathbf{H},$$

(8.8)

and

$$N_{\alpha }\left(Y\right)=1+{\displaystyle \frac{{\left|T\right|}^{p-1}-1}{2p}}\mathbf{H}.$$

(8.9)

Proof of Theorem 30. 

The automorphism $\alpha$ fixes 1 and s, and acts transitively on

$$\{r^a:1\le a\le p-1\}$$

and on

$$\{r^{a}s:1\le a\le p-1\}.$$

By Lemma 3, $N_{\alpha }\left(X\right)$ is $\alpha$-invariant, so there exist integers $x_1,x_2,x_3,x_4$ with

$$N_{\alpha }\left(X\right)=x_1·1+x_{2}s+x_3{\displaystyle \sum _{a=1}^{p-1}}{r}^a+x_4{\displaystyle \sum _{a=1}^{p-1}}{r}^{a}s.$$

(8.10)

The coefficients of X are nonnegative, and both $\alpha$ and multiplication in $\mathbb{Z}\left[D_{2p}\right]$ preserve nonnegativity; hence $x_i\in {\mathbb{Z}}_{\ge 0}$.

Put

$$u=x_1-x_3,v=x_2-x_4.$$

Using

$${\displaystyle \sum _{a=1}^{p-1}}{\zeta }^a=-1,$$

we obtain

$${\rho }_1\left(N_{\alpha }\left(X\right)\right)=\left(\begin{array}{cc}u& v\\ v& u\end{array}\right).$$

(8.11)

Let

$$\delta =\det \left({\rho }_1\left(X\right)\right).$$

By Theorem 3, $\delta$ is a real unit in $\mathbb{Z}\left[\zeta \right]$. Let q be the primitive root used in (8.1). For each i,

$$\det \left({\rho }_1\left({\alpha }^i\left(X\right)\right)\right)={\sigma }_{q^i}\left(\delta \right),$$

where ${\sigma }_t\left(\zeta \right)={\zeta }^t$. Therefore

$$\det \left({\rho }_1\left(N_{\alpha }\left(X\right)\right)\right)={\displaystyle \prod _{i=0}^{p-2}}{\sigma }_{q^i}\left(\delta \right)=N_{\mathbb{Q}\left(\zeta \right)/\mathbb{Q}}\left(\delta \right).$$

The conjugates of the real algebraic integer $\delta$ pair up as equal nonzero real numbers, ${\sigma }_t\left(\delta \right)={\sigma }_{-t}\left(\delta \right)$; hence the norm of $\delta$ is a product of positive real numbers. Being the norm of a unit, it equals $\pm 1$, and therefore

$$N_{\mathbb{Q}\left(\zeta \right)/\mathbb{Q}}\left(\delta \right)=1.$$

Consequently,

$$u^2-v^2=1.$$

Since $u,v\in \mathbb{Z}$, it follows that

$$(u,v)=(1,0)\mathrm{or}(-1,0).$$

(8.12)

On the other hand, ${\chi }_{-}$ is a ring homomorphism fixed by $\alpha$, so

$${\chi }_{-}\left(N_{\alpha }\left(X\right)\right)={\chi }_{-}{\left(X\right)}^{p-1}=\epsilon {\left(S\right)}^{p-1}=1,$$

because $\epsilon \left(S\right)\in \{-1,1\}$ and $p-1$ is even. When $v=0$, that is $x_2=x_4$,

$${\chi }_{-}\left(N_{\alpha }\left(X\right)\right)=x_1+(p-1)x_3-p{x}_4=u+p(x_3-x_4).$$

If $u=-1$, then $p(x_3-x_4)=2$, contradicting the fact that p is an odd prime. Hence

$$u=1,v=0,$$

and then $x_3=x_4$, so that

$$x_1=x_3+1,x_2=x_3=x_4.$$

Applying the augmentation map to (8.10),

$$x_1+x_2+(p-1)(x_3+x_4)=1+2p{x}_3={\left|S\right|}^{p-1}.$$

Since $\left|S\right|\left|T\right|=2pk-1$, we have $\gcd \left(\right|S|,p)=1$; together with the oddness of $\left|S\right|$, this gives $\gcd \left(\right|S|,2p)=1$. As $\phi \left(2p\right)=p-1$, Euler’s theorem yields

$$x_3={\displaystyle \frac{{\left|S\right|}^{p-1}-1}{2p}}\in {\mathbb{Z}}_{\ge 0},$$

and therefore

$$N_{\alpha }\left(X\right)=(x_3+1)·1+x_{3}s+x_3{\displaystyle \sum _{a=1}^{p-1}}{r}^a+x_3{\displaystyle \sum _{a=1}^{p-1}}{r}^{a}s=1+x_3\mathbf{H}.$$

This proves (8.8). The proof of (8.9) is identical.    □

Corollary 6 

(Four-parameter form). Under the hypotheses of Theorem 7, write

$$N_{\alpha }\left(X\right)=x_1·1+x_{2}s+x_3{\displaystyle \sum _{a=1}^{p-1}}{r}^a+x_4{\displaystyle \sum _{a=1}^{p-1}}{r}^{a}s,$$

and

$$N_{\alpha }\left(Y\right)=y_1·1+y_{2}s+y_3{\displaystyle \sum _{a=1}^{p-1}}{r}^a+y_4{\displaystyle \sum _{a=1}^{p-1}}{r}^{a}s.$$

Then the eight parameters are nonnegative integers and are uniquely determined by

$$x_1=1+{\displaystyle \frac{{\left|S\right|}^{p-1}-1}{2p}},x_2=x_3=x_4={\displaystyle \frac{{\left|S\right|}^{p-1}-1}{2p}},$$

and

$$y_1=1+{\displaystyle \frac{{\left|T\right|}^{p-1}-1}{2p}},y_2=y_3=y_4={\displaystyle \frac{{\left|T\right|}^{p-1}-1}{2p}}.$$

Proposition 3 

(Four-parameter consistency equations). Under the hypotheses of Theorem 7, the parameters above satisfy the coefficient equations

$$\begin{array}{cc}\hfill x_1+x_2+(p-1)(x_3+x_4)& ={\left|S\right|}^{p-1},\hfill \end{array}$$

(8.13)

$$\begin{array}{cc}\hfill y_1+y_2+(p-1)(y_3+y_4)& ={\left|T\right|}^{p-1},\hfill \end{array}$$

(8.14)

$$\begin{array}{cc}\hfill x_1{y}_1+x_2{y}_2+(p-1)(x_3{y}_3+x_4{y}_4)& =d+1,\hfill \end{array}$$

(8.15)

$$\begin{array}{cc}\hfill x_1{y}_2+x_2{y}_1+(p-1)(x_3{y}_4+x_4{y}_3)& =d,\hfill \end{array}$$

(8.16)

$$\begin{array}{cc}\hfill x_1{y}_3+x_2{y}_4+x_3{y}_1+x_4{y}_2+(p-2)(x_3{y}_3+x_4{y}_4)& =d,\hfill \end{array}$$

(8.17)

$$\begin{array}{cc}\hfill x_1{y}_4+x_2{y}_3+x_3{y}_2+x_4{y}_1+(p-2)(x_3{y}_4+x_4{y}_3)& =d.\hfill \end{array}$$

(8.18)

Proof of Proposition 32. 

Equations (8.13) and (8.14) are the augmentation equations used above.

For the remaining equations, use the multiplication rules

$$s^2=1,$$

(8.19)

$$s{\displaystyle \sum _{a=1}^{p-1}}{r}^a={\displaystyle \sum _{a=1}^{p-1}}{r}^{a}s,$$

(8.20)

$${\left({\displaystyle \sum _{a=1}^{p-1}}{r}^a\right)}^2=(p-1)1+(p-2){\displaystyle \sum _{a=1}^{p-1}}{r}^a,$$

(8.21)

and

$$\left({\displaystyle \sum _{a=1}^{p-1}}{r}^a\right)\left({\displaystyle \sum _{a=1}^{p-1}}{r}^{a}s\right)=(p-1)s+(p-2){\displaystyle \sum _{a=1}^{p-1}}{r}^{a}s.$$

(8.22)

Comparing coefficients in (8.6) yields (8.15)–(8.18).    □

Remark 11. 

The four-parameter system is a consistency check rather than a search problem with four free parameters: by Corollary 6, all eight parameters are determined by the explicit formulas of Theorem 7, and one verifies directly that the equations above are satisfied.

Remark 12. 

The orbit-norm formulas are conditional. In particular, the fact that α is an automorphism does not by itself permit arbitrary rearrangement of the factors in the orbit products. The stated commutativity hypothesis is sufficient to justify the rearrangement used in the proofs; we do not claim that it is necessary for the resulting norm identities.

9. A Lifting Criterion

Let

$$C_m=\mathbb{Z}/m\mathbb{Z}$$

be written additively. Let

$$L_h,M_h\subseteq C_m(h\in D_{2p})$$

be subsets. Define

$$S=\{(h,a):h\in D_{2p},a\in L_h\},$$

(9.1)

and

$$T=\{(h,b):h\in D_{2p},b\in M_h\}.$$

(9.2)

The correct criterion records the number of representations of each element rather than the elements occurring in the sumsets $L_u+M_v$; the latter lose multiplicity information (see Remark 13).

Theorem 8 

(Lifting criterion). Assume

$$\sum _{h\in D_{2p}}|L_h|\ge 2,\sum _{h\in D_{2p}}\left|M_h\right|\ge 2.$$

Then the sets S and T defined by (9.1)–(9.2) form a near-factorization of $D_{2p}\times C_m$ if and only if, for every $q\in D_{2p}$ and every $c\in C_m$,

$$\sum _{\begin{array}{c}u,v\in D_{2p}\\ uv=q\end{array}}\sum _{\begin{array}{c}a,b\in C_m\\ a+b=c\end{array}}11_{L_u}\left(a\right)11_{M_v}\left(b\right)=1-11_{\left\{1\right\}}\left(q\right)11_{\left\{0\right\}}\left(c\right).$$

(9.3)

Proof of Theorem 35. 

The left-hand side of (9.3) counts the ordered pairs

$$\left(\right(u,a),(v,b\left)\right)\in S\times T$$

whose product equals $(q,c)$, because

$$(u,a)(v,b)=(uv,a+b).$$

The right-hand side is 1 for every nonidentity element of $D_{2p}\times C_m$ and 0 for the identity. Together with the two size assumptions, this is exactly the definition of a near-factorization.    □

Corollary 7. 

Under the assumptions of Theorem 8,

$$\left|S\right|\left|T\right|=2pm-1.$$

(9.4)

Proof of Corollary 36. 

Summing the counting identity (9.3) over all $q\in D_{2p}$ and all $c\in C_m$ counts every pair in $S\times T$ exactly once, so the left-hand side sums to $\left|S\right|\left|T\right|$. The right-hand side sums to

$$(2p-1)m+(m-1)=2pm-1.$$

   □

Remark 13. 

The plain-sumset formulation

$$\underset{uv=q}{\uplus }(L_u+M_v)=\left\{\begin{array}{cc}{C}_m,\hfill & q\ne 1,\hfill \\ C_m\setminus \left\{0\right\},\hfill & q=1\hfill \end{array}\right.$$

is not sufficient: it records which elements occur in each sumset but not how many representations each element has. For example, take $m=4$, $L_1=\{0,1\}$, $L_h=⌀$ for $h\ne 1$, and $M_1=\{1,2\}$, $M_h=\{0,2\}$ for $h\ne 1$. Then each q admits exactly one nonempty sumset, with

$$L_1+M_1=\{1,2,3\}=C_4\setminus \left\{0\right\},L_1+M_q=C_4(q\ne 1),$$

so the sumset condition holds for every q. Nevertheless

$$(1,0)(1,2)=(1,1)(1,1)=(1,2)$$

gives two representations of the same element, and

$$\left|S\right|\left|T\right|=2·4p=8p\ne 8p-1=|D_{2p}\times C_4|-1.$$

Thus the sumset version is not a criterion for a near-factorization; the multiplicity information in (9.3) is essential.

10. An Exact Verification Procedure

The following proposition formalizes the computational verification of a proposed near-factorization.

Proposition 4 

(Exact verification). Let G be a finite group and let $S,T\subseteq G$ with $\left|S\right|,\left|T\right|\ge 2$. Then $(S,T)$ is a near-factorization of G if and only if the following conditions hold:

(i) 

$$\left|S\right|\left|T\right|=\left|G\right|-1;$$

(ii) 

the product map

$$\mu :S\times T⟶G,\mu (s,t)=st,$$

is injective;

(iii) 

$$1\notin \mu (S\times T).$$

Proof of Proposition 38. 

If $(S,T)$ is a near-factorization, all three conditions are immediate.

Conversely, condition (iii) implies that $im\left(\mu \right)\subseteq G\setminus \left\{1\right\}$. By (i) and (ii), $im\left(\mu \right)$ has cardinality $\left|S\right|\left|T\right|=\left|G\right|-1$, which is the cardinality of $G\setminus \left\{1\right\}$. Hence $im\left(\mu \right)=G\setminus \left\{1\right\}$, and every nonidentity element has a unique representation as a product from $S\times T$. Together with $\left|S\right|,\left|T\right|\ge 2$, this is the definition of a near-factorization.    □

For

$$G=D_{2p}\times C_m,$$

the multiplication can be implemented using

$$(r^a{s}^{\epsilon },z^i)(r^b{s}^{\delta },z^j)=\left(r^{a+{(-1)}^{\epsilon }b}{s}^{\epsilon +\delta },z^{i+j}\right),$$

(10.1)

where the exponents are reduced modulo p, 2 and m, respectively.

A GAP implementation is given below. It is assumed that S and T are lists of elements of G without repetitions.

products := [];

for s in S do

for t in T do

Add(products, s*t);

od;

od;

if Length(S) >= 2

and Length(T) >= 2

and Size(Set(products)) = Length(products)

and Set(products)

= Difference(Set(Elements(G)), [One(G)]) then

Print("Verified near-factorization\n");

else

Print("Verification failed\n");

fi;

The cardinality and injectivity checks are sufficient only when the identity is also excluded from the product set. Comparing the image with $G\setminus \left\{1\right\}$ directly, as in the code above, provides an exact check that detects errors in the representation of the group elements or the multiplication law; the size conditions $\left|S\right|,\left|T\right|\ge 2$ are part of the definition and must be checked separately.

11. Discussion and Further Questions

The results above provide three levels of information.

First, the quotient group-ring identity

$$\widehat{f}\left(S\right)\widehat{f}\left(T\right)=k\mathbf{H}-1$$

is an unconditional consequence of the near-factorization property. At the level of coefficients it is equivalent to the exact quotient coefficient system of Section 7.

Second, the irreducible representations of $D_{2p}$ yield unconditional spectral and arithmetic obstructions. The linear representation forces a precise coset imbalance of the two factors; the oddness of the two cardinalities already follows from the cardinality equation (Remark 2). The two-dimensional representations impose determinant-unit conditions in cyclotomic integer rings, which detect information invisible to the cardinality equations; the unit conditions for the various representations are Galois conjugate and hence equivalent (Remark 4). The Fourier energy identity is a consistency identity for quotient profiles: the individual lower bound it yields is already implied by the linear condition and the Cauchy–Schwarz inequality (Remark 7), whereas the joint energy inequality of Proposition 1 uses the relation between the two factors. All of these conditions are consequences of the quotient identity and may be used as preliminary filters; they are not independent of the full coefficient system.

Third, the exact quotient coefficient system provides a finite bilinear integer model. It is suitable for bounded integer feasibility, backtracking and exhaustive computation, but it is not a direct linear integer programming model; a linearization, or the strategy of fixing one array and solving for the other, would be needed for standard integer programming solvers. It is a necessary quotient-level test, but additional information is required to lift a quotient profile to actual subsets of a given extension group (Remark 9).

The orbit-norm reduction gives closed formulas for the orbit norms when the orbit-commutativity hypothesis is available. It should not, however, be interpreted as an unconditional characterization of all near-factorizations: the group ring of $D_{2p}$ is noncommutative, and the stated commutativity hypothesis is used only as a sufficient condition.

Several questions remain open. It would be interesting to determine families for which the spectral conditions and the exact quotient system are sufficient. Another problem is to classify quotient-feasible profiles for small primes p and small kernel sizes k. The lifting criterion in Theorem 8 also suggests a systematic construction method for near-factorizations of direct products and, more generally, semidirect products.