Semidirect Products, Split Exact Sequences, and all that

One of the things I’ve butted heads with in studying Lie Groups is the semidirect product and its relationship to split exact sequences. It quickly became apparent that this was a pretty sizeable hole in my basic knowledge, so I decided to clarify this stuff once and for all.

— Normal Subgroups and Quotient Groups —

First, a brief refresher on Normal subgroups and Quotient groups. We are given a group {G} and subgroup {H\subseteq G}.

  • Left cosets are written {gH} and right cosets are written {Hg}. Each is a set of elements in {G}. Not all left cosets are distinct, but any two are either equal or disjoint. Ditto for right cosets.
  • The left (right) cosets form a partition of {G}, but they do not in general form a group. We can try to imbue them with a suitable product, but there are obstructions to the group axioms. For example {g^{-1}H} is not a useful inverse since {(gh)^{-1}= h^{-1}g^{-1}}, so neither left cosets nor right cosets multiply as desired. More generally {(gg')H} does not consist of a product of an element of {gH} and an element of {g'H}.
  • We define the Quotient Set {G/H} to be the set of left cosets. As mentioned, it is not a group in general. There is an equivalent definition for right cosets, written {H\setminus{}G}, but it doesn’t appear often. In most cases we care about the two are the same.
  • It is easy to see that the condition which removes the obstruction is that {gH=Hg} for all {g}. Equivalently, {gHg^{-1}=H} for all {g}. If this holds, the cosets form a group. Often the stated condition is that the sets of left and right cosets are the same. But {g\in gH,Hg} so this is the same exact condition.
  • {H} is a Normal Subgroup if it obeys the conditions which make the cosets into a group.
  • Usually a Normal Subgroup is denoted {N}, and we write {N\triangleleft G} (or {N\trianglelefteq G}).
  • For a Normal subgroup {N}, the Quotient Set {Q=G/N} has (by definition) the natural structure of a group. It is called the Quotient Group.
  • We have two natural maps associated with a Normal Subgroup:
    • {N\xrightarrow{i} G} is an inclusion (i.e. injective), defined by {h\rightarrow h} (where the righthand {h} is viewed in {G}). This is a homomorphism defined for any subgroup, not just normal ones
    • {G\xrightarrow{q} Q} is the quotient map (surjective), defined by {g\rightarrow gN} (with the righthand viewed as a coset, i.e. an element of {G/N}). This map is defined for any subgroup, with {Q} the Quotient Set. For Normal Subgroups, it is a group homomorphism.
  • We know there is a copy of {N} in {G}. Though {Q} is derived from {G} and {N}, and possesses no new info, there may or may not be a copy of it in {G}. Two natural questions are when that is the case, and how {G}, {N}, and {Q} are related in general.

Let’s also recall the First Isomorphism Theorem for groups. Given any two groups {G} and {H} and a homomorphism {\phi:G\rightarrow H}, the following hold:

  • {\ker \phi} is a Normal Subgroup of {G}
  • {\mathop{\text{im}} \phi} is a subgroup of {H}
  • {\mathop{\text{im}} \phi} is isomorphic to the Quotient Group {G/\ker\phi}.

Again, we have to ask: since {\ker\phi} is a Normal Subgroup of {G}, and {\mathop{\text{im}}\phi} is isomorphic to the Quotient Group {G/\ker\phi} which “sort of” may have an image in {G}, is it meaningful to write something like (playing fast and loose with notation) {G\stackrel{?}{=} \ker\phi \oplus \mathop{\text{im}} \phi} (being very loose with notation)? The answer is no, it’s more complicated.

— Exact Sequences —

Next, a very brief review of exact sequences. We’ll use {1} for the trivial group. The usual convention is to use {1} for general groups and {0} for Abelian groups. An exact sequence is a sequence of homomorphisms between groups {\cdots \rightarrow G_n \xrightarrow{f_n} G_{n-1}\xrightarrow{f_{n-1}} \cdots} where {\mathop{\text{im}} f_n= \ker f_{n-1}} for every pair. Here are some basic properties:

  • {1\rightarrow A \xrightarrow{f} B\cdots} means that {f} is injective.
  • {\cdots A\xrightarrow{f} B\rightarrow 1} means that {f} is surjective.
  • {1\rightarrow A\rightarrow B\rightarrow 1} means {A=B}.
  • Short Exact Sequence (SES): This is defined as an exact sequence of the form: {1\rightarrow A\xrightarrow{f} B\xrightarrow{g} C\rightarrow 1}.
  • For an SES, {f} is injective, {g} is surjective, and {C=B/\mathop{\text{im}} f}
  • SES’s arise all the time when dealing with groups, and the critical question is whether they “split”.

We’re now ready to define Split SES’s.

  • Right Split SES: There exists a homomorphism {h:C\rightarrow B} such that {g\circ h=Id_C}. Basically, we can move to {B} and back from {C} without losing info — which means {C} is in some sense a subgroup of {B}.
  • Left Split SES: There exists a homomorphism {h:B\rightarrow A} such that {h\circ g=Id_A}. Basically, we can move to {B} and back from {A} without losing info — which means {A} is in some sense a subgroup of {B}.
  • These two conditions are not in general equal, or even equivalently restrictive. The Left Split condition is far more constraining than the Right Split one in general. The direction of the homomorphisms in the SES introduce an asymmetry. [My note: it seems likely that the two are dual in some sense.]

— External vs Internal View —

We’re going to described 3 types of group operations: the direct product, semi-direct product, and group extension. Each has a particular relationship to Normality and SES’s. There are two equivalent ways to approach this, depending whether we prefer to define a binary operation between two distinct groups or to consider the relationship amongst subgroups of a given group.

  • External view: We define a binary operation on two distinct, unrelated groups. Two groups go in, and another group comes out.
  • Internal view: We define a relationship between a group and various groups derived from it (ex. Normal or Quotient).
  • These approaches are equivalent. The Internal view describes the relationship amongst the two groups involved in the External view and their issue. Conversely, the derived groups in the Internal view may be recombined via the External view operation.

We must be a little careful with notation and terminology. When we use the symbol {HK}, it can mean one of two things.

  • Case 1: {H} and {K} are distinct groups. {HK} is just the set of all pairs of elements {(h,k)}. I.e. it is the direct product set (but not group).
  • Case 2: {H} and {K} are subgroups of a common group {G} (or have some natural implicit isomorphisms to such subgroups). In this case, {HK} is the set of all elements in {G} obtained as a product of an element of {H} and an element of {K} under the group multiplication.
  • Note that we may prefer cases where two subgroups cover {G}, but there are plenty of other possibilities. For example, consider {Z_{30}} (the integers mod 30). This has several obvious subgroups ({Z_2}, {Z_3}, {Z_5}, {Z_6}, {Z_{10}}, {Z_{15}}). {Z_2} and {Z_3} only intersect on {0} (the additive identity). However, the two do not cover (or even generate) the group! Similarly, {Z_2} and {Z_{10}} do not cover the group (or even generate it) but intersect on a nontrivial subset!
  • Going the other way, we’ll say that {G=HK} if {H} and {K} are subgroups and every element {g} can be written as {hk} for some {h\in H} and {k\in K}. Note that {H} and {K} need not be disjoint (or even cover {G} set-wise).

Another potentially confusing point should be touched on. When we speak of “disjoint” subgroups {H} and {K} we mean that {H\cap K=\{e\}}, NOT that it is the null set. I.e., {H\cap K= 1}, the trivial group.

— Semidirect Product —

The semidirect product may seem a bit arbitrary at first but, as we will see, it is a natural part of a progression which begins with the Direct Product. Here are the two ways of defining it.

  • External view (aka Outer Semidirect Product): Given two groups {H} and {K} and a map {\phi:K\rightarrow Aut(H)}, we define a new group {H\rtimes K}. We’ll denote by {\phi_k(h)} the effect of the automorphism {\phi(k)} on {h} (and thus an element of {H}). Set-wise, {H\rtimes K} is just {H\times K} (i.e. all pairs {(h,k)}). The identity is {(e,e)}. Multiplication on {H\rtimes K} is defined as {(h,k)(h',k')= (h\phi_k(h'),kk')}. The inverse is {(h,k)^{-1}= (\phi_{k^{-1}}(h^{-1}),k^{-1})}.
  • Internal view (aka Inner Semidirect Product): Given a group {G} and two disjoint subgroups {N} and {K}, such that {G=NK} and {N} is a Normal Subgroup, {G} is called the Semidirect product {N\rtimes K}. The normality of {H} constrains {K} to be isomorphic to the Quotient Group {G/N}.

There are a few important things to note about this.

  • There are (potentially) many Semidirect products of two given groups, obtained via different choices of {\phi}. The notation is deceptive because it hides our choice of {\phi}. Given any {H,K,\phi} there exists a Semidirect product {H\rtimes K}. The various Semidirect products may be isomorphic to one another, but in general need not be. I.e., a given {H} and {K} may have multiple distinct semidirect products. This actually happens. Wikipedia mentions that there are 4 non-isomorphic semidirect products of {C_8} and {C_2} (the former being the Normal Subgroup in each case). One is a Direct Product, and the other 3 are not.
  • It also is possible for a given group {G} to arise from several distinct Semidirect products (of different pairs of groups). Again from Wikipedia, there is a group of order 24 which can be written as 4 distinct semiproducts of groups.
  • Yet another oddity is that a seemingly nontrivial {H\rtimes K} can be isomorphic to {H\oplus K}.
  • If {\phi= Id} (i.e. every {k} maps to the identify map on {H}), then {G=H\oplus K}.
  • To go from the External view to the Internal one, we note that, by construction, {H} is a Normal Subgroup of {H\rtimes K} and {K} is the Quotient Group {G/H}. To be precise, the Normal Subgroup is {(N,e)}, which is isomorphic to {N}, and the Quotient Group {G/(N,e)} is isomorphic to {K}.
  • To go from the Internal view to the External one, we choose {\phi_k(h)= khk^{-1}} as our function. I.e., {\phi} is just conjugation by the relevant element.
  • It may seem like there is an imbalance here. For a specific choice of Normal Subgroup {N}, the External view offers complete freedom of {\phi}, while the Internal view has a fixed {\phi}. Surely the latter is a special case of the former. The fallacy in this is that we must consider the pair {(G,N)}. We very well could have non-isomorphic {G,G'} with Normal Subgroups {N,N'} where {N\approx N'}. I.e. they are the same Normal Subgroup, but with different parent groups. We then would have different {\phi}‘s via our Internal view procedure. The correspondence is between {(H,K,\phi)} and {(G,N,K)} choices. Put differently, the freedom in {\phi} loosely corresponds to a freedom in {G}.
  • Note that, given {G} and a Normal Subgroup {N} — with the automatic Quotient Group {G/N} — we do NOT necessarily have a Semidirect product relationship. The condition of the Semidirect product is stricter than this. As we will see it requires not just isomorphism, but a specific isomorphism, between {H} and {G/N}. Equivalently, it requires a Right-Split SES (as we will discuss).
  • The multiplication defined in the External view may seem very strange and unintuitive. In essence, here is what’s happening. For a direct product, {H} and {K} are independent of one another. Each half of the pair acts only on its own elements. For a semidirect product, the non-normal half {K} can twist the normal half {H}. Each element of {K} can alter {H} in some prescribed fashion, embodied in {\phi(k)}. So {K} is unaffected by {H} but {H} can be twisted by {K}.
  • It is interesting to compare the basic idea to that of a Fiber bundle. There, the fiber can twist (via a group of homeomorphisms) as we move around the base space. Here, the normal subgroup can twist as we move around the non-normal part. Each generalizes a direct product and measures our need to depart from it.
  • The semidirect product of two groups is Abelian iff it’s just a direct product of abelian groups.

— Group Extensions —

As with Semidirect products, there are 2 ways to view these. To make matters confusing, the notation speaks to an Internal view, while the term “extension” speaks to an External view.

  • External view: Given groups {A} and {C}, we say that {B} is an extension of {C} by {A} if there is a SES {1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1}.
  • Internal view: Given a group {G} and Normal Subgroup {N\triangleleft G}, we say that {G} is an extension of {Q} by {N}, where {Q=G/N} is the Quotient Group.
  • Note that the two are equivalent. If {B} is an extension of {A} by {C}, then {A} is Normal in {B} and {C} is isomorphic to the Quotient Group {B/A}.
  • Put simply, the most general form of the Group, Normal Subgroup, induced Quotient Group trio is the Group Extension.

— Direct Products, Semidirect Products, and Group Extensions —

In the External view, we’ve mentioned three means of getting a group {B} from two groups {A} and {C}:

  • Direct Product: {B=A\oplus C}. This is unique.
  • Semidirect Product: {B=A\rtimes C}. There may multiple of these, corresponding to different {\phi}‘s.
  • Group Extension: A group {B} for which there are 2 homomorphisms forming a SES {1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1}. There may be many of these, corresponding to different choices of the two homomorphisms.

Equivalently, we have several ways of describing the relationship between two subgroups {H,K\subseteq G} which are disjoint (i.e. {H\cap K=\{e\}}).

  • Direct Product: {G=H\oplus K} requires that both be Normal Subgroups.
  • Semidirect Product: {G=H\rtimes K} requires that {H} be normal (in which case, {Q=G/H}, and {\phi} is determined by it). For a given {H} there may be multiple, corresponding to different {G}‘s.
  • Group Extension: Both {H} and {K} sit in {G} to some extent. {H} must be Normal.

Note that not every possible relationship amongst groups is captured by these. For example, we could have two non-normal subgroups or two homomorphisms which don’t form an SES, or no relationship at all.

An excellent hierarchy of conditions was provided by Arturo Magidin in answer to someone’s question on Stackoverflow. I roughly replicate it here. Unlike him, I’ll be sloppy and not distinguish between subgroups and groups isomorphic to subgroups.

  • Direct Product ({G=H\oplus K}): {H,K} both Normal Subgroups. {H,K} disjoint. {G=HK}
  • Semidirect Products ({G=H\rtimes K}): {H} Normal Subgroup, {K} Subgroup. {H,K} disjoint. {G=HK}. I.e., we lose Normality of {K}.
  • Group Extension ({G} is extension of {H} by {K}): {H} Normal Subgroup, {G/H\approx K}. I.e. {K} remains the Quotient Group (as before), but the Quotient Group may no longer be a subgroup of {G} at all!

Now is a good time to mention the relationship between the various SES Splitting conditions:

  • For all groups: Left Split is equivalent to {B=A\oplus C}, and they imply Right Split. (LS=DP) => RS always.
  • For abelian groups, the converse holds and Right split implies Left Split and Direct Sum. I.e. the conditions are equivalent. LS=DP=RS for Abelian.
  • For nonabelian groups: Right Split implies {B=A\rtimes C} (with {\phi} depending on the SES map). We’ll discuss this shortly.

Back to the hierarchy, now from a SES standpoint:

  • Most general case: There is no SES at all. Given groups {A,B,C}, there may be no homomorphisms between them. If there are homomorphisms, there may be none which form an SES. Consider a general pair of homomorphisms {f:A\rightarrow B} and {g:B\rightarrow C}, with no assumptions. We may turn to the first isomorphism theorem for help, but that does us no good. The first isomorphism theorem says that {\ker f \triangleleft B} and {\mathop{\text{im}} f\approx A/\ker f}, and {\ker g \triangleleft C} and {\mathop{\text{im}} g\approx B/\ker g}. This places no constraints on {A} or {C}.
  • Group Extension: Any SES defines a group extension. They are the same thing.
  • Semidirect Product: Any SES which right-splits corresponds to a Semidirect Product (with the right-split map determining {\phi})
  • Direct Product: Any SES which left-splits (and thus right-splits too) corresponds to a direct product.

So, when we see the standard SES: {1\rightarrow N\rightarrow G\rightarrow G/N\rightarrow 1}, this is a group extension. Only if it right splits can we write {G= N\rtimes G/N}, and only if it left splits can we write {G= N\oplus G/N}.

— Some Notes —

  • Group Extensions are said to be equivalent if their {B}‘s are isomorphic and there exists an isomorphism between them which makes a diamond diagram commute. It is perfectly possible for the {B}‘s to be isomorphic but for two SES’s not to be equivalent extensions.
  • Subtlety referred to above. A quotient group need not be isomorphic to a subgroup of {G}. It only is defined when {N} is normal, and there automatically is a surjective homomorphism {G\rightarrow Q}. But we don’t have an injective homomorphism {Q\rightarrow G}, which is what would be need for it to be isomorphic to a subgroup of {G}. This is precisely what the right-split furnishes. In that case, it is indeed a subgroup of {G}. The semidirect product may be thought of as the statement that {Q} is a subgroup of {G}.
  • In the definition of right split and left split, the crucial aspect of the “inverse” maps is that they be homomorphisms. A simple injective (for right-split, or surjective for left-split) map is not enough!
  • It is sometimes said that the concept of subgroup is dual to the concept of quotient group. This is intuitive in the following sense. A subgroup can be thought of as an injective homomorphism. By the SES for normal/quotient groups, we can think of a quotient group as a surjective homomorphism. Since injections and surjections are categorically dual, it makes sense to think of quotient groups and subgroups as similarly dual. Whether the more useful duality is subgroup quotient group or normal subgroup quotient group is unclear to me.

180 Women and Sun Tzu

It is related that Sun Tzu (the elder) of Ch’i was granted an audience with Ho Lu, the King Of Wu, after writing for him a modest monograph which later came to be known as The Art of War. A mere scholar until then (or as much of a theorist as one could be in those volatile times), Sun Tzu clearly aspired to military command.

During the interview, Ho Lu asked whether he could put into practice the military principles he expounded — but using women. Sun Tzu agreed to the test, and 180 palace ladies were summoned. These were divided by him into two companies, with one of the King’s favorite concubines given the command of each.

Sun Tzu ordered his new army to perform a right turn in unison, but was met with a chorus of giggles. He then explained that, “If words of command are not clear and distinct, if orders are not thoroughly understood, then the general is to blame.” He repeated the order, now with a left turn, and the result was the same. He now announced that, “If words of command are not clear and distinct, if orders are not thoroughly understood, then the general is to blame. But if his orders are clear, and the soldiers nevertheless disobey, then it is the fault of their officers,” and promptly ordered the two concubines beheaded.

At this point, Ho Lu intervened and sent down an order to spare the concubines for he would be bereft by their deaths. Sun Tzu replied that, “Having once received His Majesty’s commission to be the general of his forces, there are certain commands of His Majesty which, acting in that capacity, I am unable to accept.” He went ahead and beheaded the two women, promoting others to fill their commands. Subsequent orders were obeyed instantly and silently by the army of women.

Ho Lu was despondent and showed no further interest in the proceedings, for which Sun Tzu rebuked him as a man of words and not deeds. Later he was commissioned a real general by Ho Lu, proceeded to embark on a brilliant campaign of conquest, and is described as eventually “sharing in the might of the king.”

This is a particularly bewildering if unpleasant episode. Putting aside any impression the story may make on modern sensibilities, there are some glaring incongruities. What makes it more indecipherable still is that this is the only reputable tale of Sun Tzu the elder. Apart from this and the words in his book, we know nothing of the man, and therefore cannot place the event in any meaningful context. Let us suppose the specifics of the story are true, and leave speculation on that account to historians. The episode itself raises some very interesting questions about both Sun Tzu and Ho Lu.

It is clear that Sun Tzu knew he would have to execute the King’s two favorite concubines. The only question is whether he knew this before he set out for the interview or only when he acceded to the King’s request. Though according to the tale it was Ho Lu who proposed a drill with the palace women, Sun Tzu must have understood he would have to kill not just two women but these specific women.

Let’s address the broader case first. It was not only natural but inevitable that court ladies would respond to such a summons in precisely the manner they did. Even if we ignore the security they certainly felt in their rank and the affections of the King, the culture demanded it. Earnest participation in such a drill would be deemed unladylike. It would be unfair to think the court ladies silly or foolish. It is reasonable to assume that in their own domain of activity they exhibited the same range of competence and expertise as men did in martial affairs. But their lives were governed by ceremony, and many behaviours were proscribed. There could be no doubt they would view the proceedings as a game and nothing more. Even if they wished to, they could not engage in a serious military drill and behave like men without inviting quiet censure. The penetrating Sun Tzu could not have been unaware of this.

Thus he knew that the commanders would be executed. He may not have entered the King’s presence expecting to kill innocent women, but he clearly was prepared to do so once Ho Lu made his proposal. In fact, Sun Tzu had little choice at that point. Even if the King’s proposal was intended in jest, he still would be judged by the result. Any appearance of frivolity belied the critical proof demanded of him. Sun Tzu’s own fate was in the balance. He would not have been killed, but he likely would have been dismissed, disgraced, and his ambitions irredeemably undermined.

Though the story makes the proposal sound like the whimsical fancy of a King, it very well could have been a considered attempt to dismiss a noisome applicant. Simply refusing an audience could have been impolitic. The man’s connections or family rank may have demanded suitable consideration, or perhaps the king wished to maintain the appearance of munificence. Either way, it is plausible that he deliberately set Sun Tzu an impossible task to be rid of him without the drawbacks of a refusal. The King may not have known what manner of man he dealt with, simply assuming he would be deterred once he encountered the palace ladies.

Or he may have intended it as a true test. One of the central themes of Chinese literature is that the monarch’s will is inviolable. Injustice or folly arises not from a failing in the King but from venal advisers who hide the truth and misguide him. A dutiful subject seeks not to censure or overthrow, but rather remove the putrescence which clouds the King’s judgment with falsehood, and install wise and virtuous advisers. Put simply, the nature of royalty is virtuous but it is bound by the veil of mortality, and thus can be deceived. One consequence of this is that disobedience is a sin, even in service of justice. Any command must be obeyed, however impossible. This is no different from Greek mythology and its treatment of the gods. There, the impossible tasks often only could be accomplished with magical assistance. In Sun Tzu’s case, no magic was needed. Only the will to murder two great ladies.

As for the choice of women to execute, it does not matter whether the King or Sun Tzu chose the disposition of troops and commands. The moment Sun Tzu agreed to the proposal, he knew not only that he would have to execute women but which ones. Since he chose, this decision was made directly. But even if it had been left to the king, there could be no question who would be placed in command and thus executed.

The palace hierarchy was very strict. While the ladies probably weren’t the violent rivals oft depicted in fiction, proximity to the King — or, more precisely, place in his affections, particularly as secured by production of a potential heir — lent rank. No doubt there also was a system of seniority based on age and family, among the women, many of whom probably were neither concubines nor courtesans, but noble-women whose husbands served the King. It was common for ladies to advance their husbands’ (and their own) fortunes through friendship with the King’s concubines. Whatever the precise composition of the group, a strict pecking order existed. At the top of this order were the King’s favorites. There could be no other choice consistent with domestic accord and the rules of precedence. Those two favorite concubines were the only possible commanders of the companies.

To make matters worse, those concubines may already have produced heirs. Possibly they were with child at that very moment. This too must have been clear to Sun Tzu. Thus he knew that he must kill the two most beloved of the King’s concubines, among the most respected and noblest ladies in the land, and possibly the mothers of his children. Sun Tzu even knew he may be aborting potential heirs to the throne. All this is clear as day, and it is impossible to imagine that the man who wrote the Art of War would not immediately discern it.

But there is something even more perplexing in the story. The King did not stop the executions. Though the entire affair took place in his own palace, he did not order his men to intervene, or even belay Sun Tzu’s order. He did not have Sun Tzu arrested, expelled, or executed. Nor did he after the fact. Ho Lu simply lamented his loss, and later hired the man who had effected it.

There are several explanations that come to mind. The simplest is that he indeed was a man of words and not deeds, cowed by the sheer impetuosity of the man before him. However, subsequent events do not support this. Such a man would not engage in aggressive wars of conquest against his neighbors, nor hire the very general who had humiliated and aggrieved him so. Perhaps he feared that Sun Tzu would serve another, turning that prodigious talent against Wu. It would be an understandable concern for a weak ruler who dreaded meeting such a man on the battlefield. But it also was a concern which easily could have been addressed by executing him on the spot. The temperamental Kings of fable certainly would have. Nor did Ho Lu appear to merely dissemble, only to visit some terrible vengeance on the man at a later date. Sun Tzu eventually became his most trusted adviser, described as nearly coequal in power.

It is possible that Ho Lu lacked the power oft conflated with regality, and less commonly attendant upon it. The title of King at the time meant something very different from modern popular imaginings. The event in question took place around 500 BC, well before Qin Shi Huang unified China — briefly — with his final conquest of Qi in 221 BC. In Ho Lu’s time, kingdoms were akin to city-states, and the Kings little more than feudal barons. As in most historical treatises, troop numbers were vastly exaggerated, and 100,000 troops probably translated to a real army of mere thousands.

This said, it seems exceedingly improbable that Ho Lu lacked even the semblance of authority in his own palace. Surely he could execute or countermand Sun Tzu. Nor would there be loss of face in doing so, as the entire exercise could be cast as farcical. Who would object to a King stopping a madman who wanted to murder palace concubines? If Sun Tzu was from a prominent family or widely regarded in his own right (which there is no evidence for), harming him would not have been without consequence. But there is a large difference between executing the man and allowing him to have his way in such a matter. Ho Lu certainly could have dismissed Sun Tzu or proposed a more suitable test using peasants or real soldiers. To imagine that a king would allow his favorite concubines to be executed, contenting himself with a feeble protest, is ludicrous. Nor was Sun Tzu at that point a formidable military figure. A renowned strategist would not have troubled to write an entire treatise just to impress a single potential patron. That is not the action of a man who holds the balance of power.

The conclusion we must draw is that the “favorite concubines” were quite dispensible, and the King’s protest simply the form demanded by propriety. He hardly could not protest the murder of two palace ladies. Most likely, he used Sun Tzu to rid himself of two problems. At the very least, he showed a marked lack of concern for the well-being of his concubines. We can safely assume that his meat and drink did not lose their savour, as he envisioned in his tepid missive before watching Sun Tzu behead the women.

While it is quite possible that he believed Sun Tzu was just making a point and would stop short of the actual execution, this too seems unlikely. The man had just refused a direct order from the King, and unless the entire matter was a tremendous miscommunication there could be little doubt he would not be restrained.

Ho Lu may genuinely have been curious to see the outcome. Even he probably could not command obedience from the palace ladies, and he may have wished to see what Sun Tzu could accomplish. But more than this, the King probably felt Sun Tzu was a valuable potential asset. The matter then takes on a very different aspect.

From this viewpoint, Ho Lu was not the fool he seemed. The test was proposed not in jest, but in deadly earnest, and things went exactly as he had hoped but not expected. He may have had to play the indolent monarch, taking nothing seriously and bereaved by a horrid jest gone awry. It is likely he was engaging in precisely the sort of deception Sun Tzu advocated in his treatise. He appeared weak and foolish, but knew exactly what he wanted and how to obtain it.

This probably was not lost on Sun Tzu, either. Despite his parting admonition, he did later agree to serve Ho Lu. It is quite possible that the king understood precisely the position he was placing Sun Tzu in, and anticipated the possible executions. Even so, he may have been uncertain of the man’s practical talent and the extent of his will. There is a great divide between those who write words and those who heed them. Some may bridge it, most do not. Only in the event did Sun Tzu prove himself.

For this reason, Ho Lu could not be certain of the fate of the women. Nonetheless he placed them in peril. They were disposable, if not to be disposed of. It seems plausible that an apparently frivolous court game actually was a determined contest between two indomitable wills. The only ones who did not grasp this, who could not even recognize the battlefield on which they stepped solely to shed blood, were the concubines.

By this hypothesis, they were regarded as little more than favorite dogs or horses, or perhaps ones which had grown old and tiresome. A King asks an archer to prove his skill by hitting a “best” hound, then sets the dog after a hare, as he has countless times before. The dog quickens to the chase, eagerly performing as always, confident that its master’s love is timeless and true. Of all present, only the dog does not know it is to be sacrificed, to take an arrow to prove something which may or may not be of use one day to its master. If the arrow falls short, it return to its master’s side none the wiser and not one jot less sure of its place in the world or secure in the love of its master, until another day and another archer. This analogy may seem degrading and insulting to the memory of the two ladies, but that does not mean it is inaccurate. It would be foolhardy not to attribute such views to an ancient King and general simply because we do not share them or are horrified by them or wish they weren’t so. In that time and place, the concubines’ lives were nothing more than parchment. The means by which Ho Lu and Sun Tzu communicated, deadly but pure.

The view that Ho Lu was neither a fool nor a bon vivant is lent credence by the manner of his rise to power. He usurped the throne from his uncle, employing an assassin to accomplish the task. This and his subsequent campaign of conquest are not the actions of a dissipated monarch. Nor was he absent from the action, wallowing in luxury back home. In fact, Ho Lu died from a battle wound during his attempted conquest of Yue.

It is of course possible that the true person behind all these moves was Wu Zixu, the King’s main advisor. But by that token, it also is quite possible that the entire exercise was engineered by Wu Zixu — with precisely intended consequences, perhaps ridding himself of two noisome rivals with influence over the King. In that case, the affair would be nothing more than a routine palace assassination.

Whatever the explanation, we should not regard the deaths of the two concubines as a pointless tragedy. The discipline instilled by two deaths could spare an entire army from annihilation on the field. Sun Tzu posited that discipline was one of the key determinants of victory, and in this he was not mistaken. That is no excuse, but history needs none. It simply is.

This said, it certainly is tempting to regard the fate of these ladies as an unadorned loss. Who can read this story and feel anything but sadness for the victims? Who can think Sun Tzu anything but a callous murderer, Ho Lu anything but foolish or complicit? It is easy to imagine the two court concubines looking forward to an evening meal, to poetry with their friends, to time with their beloved husband. They had plans and thoughts, certainly dreams, and perhaps children they left behind. One moment they were invited to play an amusing game, the next a sharp metal blade cut away all they were, while the man they imagined loved them sat idly by though it lay well within his power to save them. Who would not feel commingled sorrow and anger at such a thing? But that is not all that happened.

A great General was discovered that day, one who would take many lives and save many lives. Whether this was for good or ill is pointless to ask and impossible to know. All we can say is that greatness was achieved. 2500 years later and in a distant land we read both his tale and his treatise.

Perhaps those two died not in service to the ambition of one small general in one small kingdom. Perhaps they died so centuries later Cao Cao would, using the principles in Sun Tzu’s book, create a foundation for the eventual unification of China. Or so that many more centuries later a man named Mao would claim spiritual kinship and murder a hundred million to effect a misguided economic policy. Would fewer or more have died if these two women had lived? Would one have given birth to a world-conquering general, or written a romance for the ages?

None of these things. They died like everyone else — because they were born. The axe that felled them was wielded by one man, ordered by another, and sanctioned by a third. Another made it, and yet another dug the ore. Are they all to blame? The affair was one random happening in an infinitude of them, neither better nor worse. A rock rolls one way, but we do not condemn. It rolls another, but we do not praise.

But we do like stories, and this makes a good one.

[Source: The account itself is taken from The Art of War with Commentary, Canterbury Classics edition, which recounted it from a translation of the original in the Records of the Grand Historian. Any wild speculation, ridiculous hypotheses, or rampant mischaracterizations are my own.]