The Abelian group criterion for a product of two prime squares applies because $11<19$ and $11\nmid 19^2-1=360$. Here its Sylow theorems proof is especially short:The candidates for $n_{19}$ are $1,11,121$, of which only $1$ has the required remainder. The candidates for $n_{11}$ are $1,19,361$, again leaving only $1$. Thus both Sylow subgroups $P,Q$ are normal subgroups. Their group commutators lie in $P\cap Q=\{1\}$, and $PQ=G$, so $G\cong P\times Q$. Both factors are abelian groups of prime-square order by (a).

The Fundamental theorem of finitely generated abelian groups gives preciselyThese four groups are pairwise nonisomorphic: their characteristic Sylow subgroups differ in whether they contain an element of order $11^2$ or $19^2$. The first is also the cyclic group $C_{43681}$.

Write $|G|=pq$ with $p<q$. The Sylow theorems give $n_q\mid p$ and $n_q\equiv 1(\mathrm{mod}q)$. Since $p<q$, necessarily $n_q=1$. Its unique Sylow subgroup $N$ is a normal subgroup and has prime order $q$, soA Sylow subgroup $H\cong C_p$ has trivial intersection with $N$, and $|NH|=|N||H|=pq$, so $NH=G$. ThusThe semidirect product action is a group homomorphism $C_p\to \mathrm{Aut}(C_q)\cong C_{q-1}$. If the action is trivial, $G\cong C_{pq}$ by the Chinese remainder theorem; if it is nontrivial, $p\mid q-1$. In particular $G$ has both a cyclic normal subgroup and a cyclic quotient, making it a metacyclic group and hence a solvable group.

Let $|G|=p^2$. The allowed center of a group theorem gives $|Z(G)|>1$. By Lagrange's theorem its order is $p$ or $p^2$. In the first case $G/Z(G)$ has order $p$ and is a cyclic group. A cyclic quotient by the center forces $G$ to be Abelian: if $G/Z(G)=⟨gZ(G)⟩$, write arbitrary elements as $g^{a}z,g^{b}w$ with $z,w\in Z(G)$, and commute them directly. Hence in every caseFor two examples at prime-cube order, take the dihedral group of a square, $⟨r,s\mid r^4=s^2=1,srs=r^{-1}⟩$, and the quaternion group $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$. Both have order $8=2^3$. They are nonabelian because $sr\ne rs$ and $ij=k\ne -k=ji$, respectively. They are distinct examples even up to group isomorphism: the dihedral group has five elements of order two and the quaternion group has only one. The request asks for two examples; it does not require the prime to be arbitrary.

Complete the square to obtain the equivalent quadratic congruenceHere $\gcd (1111,8093)=1$: the Euclidean algorithm has successive nonzero remainders $316,163,153,10,3,1$. If this modular congruence had a solution, the Jacobi symbol $(1111/8093)$ would be $1$, since a square coprime to an odd modulus has Legendre symbol $1$ at every prime factor. Calculate instead, using Jacobi reciprocity and reduction of numerators,Therefore there is no solution. This obstruction does not require a separate primality test for $8093$; the Jacobi symbol suffices.

Write $P={\prod }_i{p}_i$, $Q={\prod }_j{q}_j$, counting repetitions. Since $P,Q$ are coprime, every pair $p_i,q_j$ consists of distinct odd prime numbers. Quadratic reciprocity for the Legendre symbol givesMultiplying over all $i,j$ and using multiplicativity of the Jacobi symbol yields the sign exponentby the odd-product parity identity proved in (b)(i). Thus the Jacobi reciprocity law isIf either integer is $1$, its factorization is empty and both sides equal $1$.

The visual system must infer relatively stable object properties from images jointly determined by illumination, shape, material and viewpoint. The problem is intrinsically ambiguous; robust performance exploits several constraints and tolerates uncertainty rather than perfectly recovering a unique physical scene from every image.

Even a diffuse model contains a surface-illuminant ambiguity. Ignoring a fixed geometric factor, $N(\lambda )=E(\lambda )R(\lambda )$. For any positive function $g$ that leaves the resulting reflectance physically admissible, define $E^{\prime }=Eg$ and $R^{\prime }=R/g$. Then $E^{\prime }{R}^{\prime }=ER$, so even an observer measuring the complete outgoing spectrum could not distinguish those two explanations from that surface alone. Three cone catches introduce further spectral metamerism. Thus perfect color constancy cannot be obtained by a universal local inversion of three numbers.

For directional optical reflection, an image sample instead depends on an integral such asThe expression sums contributions from incident directions, weighted by the bidirectional reflectance distribution function and projected area. It assumes a local, nonfluorescent optical reflection model and omits emission. A glossy surface therefore reflects aspects of its environment, and moving the observer changes the sampled outgoing direction even if the material is unchanged. Subsurface transport and transmission require further terms or models.

The system can constrain illumination by comparing surfaces, spatial boundaries and changes over time, while using shape and scene layout to distinguish a shadow from a pigment boundary. Cone-gain adjustment can approximately compensate shared spectral changes but cannot solve every mixture of differently colored lights, interreflections and specular components. A spatial average is unreliable if a scene is genuinely dominated by one colored material. A bright region might be a lamp, a highlight or a high-reflectance object, so “assume the brightest point is white” is not a universal solution.

Material judgments require distinguishing diffuse shading from highlights and transparency or translucency cues. Isamu Motoyoshi and colleagues showed that manipulating image statistics, including luminance-distribution skewness, could influence apparent gloss and lightness in their surface-image conditions.^[ref] Such regularities offer efficient cues. They cannot uniquely specify a material: a histogram-preserving spatial permutation changes spatial organization while retaining the luminance histogram and all its moments. That mathematical ambiguity motivates experiments separating histogram cues from geometric consistency; it does not, by itself, predict every appearance judgment after scrambling.

Juno Kim and colleagues directly tested the relation between highlights and diffuse shading. Manipulating their positional and orientation congruence reduced perceived gloss, including when simple proximity to broad luminance maxima was controlled.^[ref] The visual system therefore uses relationships among image structures, not merely the presence of a few bright pixels. The conclusion concerns the tested images and judgments, rather than a claim that one cue explains every shiny or transparent material.

Temporal information and active viewing help further. A highlight can move differently from a surface marking; multiple views constrain geometry, while familiar object structure and expected illumination supply priors. These strategies can fail under unusual lighting or deliberately misleading images. The system must also decide which differences matter: recognizing a material category, judging gloss and estimating spectral reflectance are related but separable goals.

The research challenge is to retain causal control while measuring appearance under the coupled spectral, spatial and temporal variations of actual scenes. Simply replacing a flat patch with a photograph does not guarantee that either the stimulus or the task is ecologically informative.

A display's RGB values do not specify the spectrum reaching a participant's cone cells. Primary spectra, viewing geometry, calibration, luminance range and the observer's lens and pigment sensitivities matter. A photograph can clip highlights, alter white balance and compress dynamic range; a display cannot necessarily reproduce the original spectrum, brightness or angular light field. A researcher studying spectral constancy should measure radiance and illuminants or specify the limits of a display simulation, rather than treating an uncalibrated image as physical ground truth.

Real materials also vary in more than diffuse reflectance. Shape, roughness, specularity, translucency and surrounding illumination jointly determine an image. Changing one rendering parameter can unintentionally change highlight area, contrast and apparent shape. Conversely, holding an image statistic constant can produce an unnatural stimulus. Experiments need independent manipulation of candidate causes and a clear statement of which physical scene properties are represented by the apparatus.

James M. Kraft and David H. Brainard demonstrated a productive compromise with a chamber of actual surfaces. They independently controlled illumination and surface reflectances and used a projection colorimeter to let observers adjust a test patch to appear achromatic. Manipulations could hold candidate image statistics approximately fixed while changing the illumination. The results rejected explanations based solely on local adaptation, the image mean or the most intense region.^[ref] The achievement was not realism alone: it was enough realism combined with manipulations that made competing mechanisms disagree. These results do not show that local adaptation never contributes.

For material appearance, Roland W. Fleming and colleagues used rendered objects illuminated by natural illumination distributions and systematically altered those distributions. Reflectance matching was more reliable under realistic illumination statistics than under some altered statistics.^[ref] This makes illumination an experimental variable, not merely a nuisance background. It also warns that a model tested only under a small set of studio lights can exploit cues unavailable or misleading elsewhere.

The dependent variable needs equal care. Selecting the same paint, matching apparent color, adjusting an achromatic point, detecting an illuminant change and naming a material do not measure the same outcome. A person may correctly infer that two patches are the same material while still noticing an appearance difference. Instructions, response format, adaptation time and available context can alter the task. Eye and head movements change retinal exposure, so fixation constraints should be chosen deliberately and their cost to natural viewing acknowledged.

A strong program therefore combines calibrated simplified stimuli, controlled real scenes, validated rendering and tests on previously unseen objects and illumination. It should sample more than one observer and record relevant visual variation, report both accuracy and uncertainty, and test model generalization across scenes rather than only across random frames of the same scene. Proposed interventions can vary geometry while preserving spectral input, or alter illumination while suppressing a particular cue. Their outcomes must be measured, not assumed. The goal is interpretable generalization: experimental control sufficient to distinguish mechanisms and scene diversity sufficient to test whether those mechanisms remain useful.