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# Category: groups

Here’s a nice, symmetric, labeled graph:

The prime numbers labelling the vertices are exactly the prime divisors of the order of the largest sporadic group: the monster group $\mathbb{M}$.
$\# \mathbb{M} = 2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71$

Looking (for example) at the character table of the monster you can check that there is an edge between two primes $p$ and $q$ exactly when the monster has an element of order $p.q$.

Now the fun part: this graph characterises the Monster!

There is no other group $G$ having only elements of these prime orders, and only these edges for its elements of order $p.q$.

This was proved by Melissa Lee and Tomasz Popiel in $\mathbb{M}, \mathbb{B}$, and $\mathbf{Co}_1$ are recognisable by their prime graphs, by using modular character theory.

The proof for the Monster takes less than one page, so it’s clear that it builds on lots of previous results.

There’s the work of Mina Hagie The prime graph of a sporadic simple group, who used the classification of all finite simple groups to put heavy restrictions on possible groups $G$ having the same prime graph as a sporadic simple group.

For the Monster, she proved that if the prime graph of $G$ is that of the monster, then the Fitting subgroup $F(G)$ must be a $3$-group, and $G/F(G) \simeq \mathbb{M}$.

Her result, in turn, builds on the Gruenberg-Kegel theorem, after Karl Gruenberg and Otto Kegel.

The Gruenberg-Kegel theorem, which they never published (a write-up is in the paper Prime graph components of finite groups by Williams), shows the wealth of information contained in the prime graph of a finite group. For this reason, the prime graph is often called the Gruenberg-Kegel graph.

The pictures above are taken from a talk by Peter Cameron, The Gruenberg-Kegel graph. Peter Cameron’s blog is an excellent source of information for all things relating groups and graphs.

The full proof of the Gruenberg-Kegel theorem is way too involved for a blogpost, but I should give you at least an idea of it, and of one of the recurrent tools involved, the structural results on Frobenius groups by John Thompson.

Here’s lemma 1.1 of the paper On connection between the structure of a finite group and the properties of its prime graph by A.V. Vasil’ev.

Lemma: If $1 \triangleleft K \triangleleft H \triangleleft G$ is a series of normal subgroups, and if we have primes $p$ dividing the order of $K$, $q$ dividing the order of $H/K$, and $r$ dividing the order of $G/H$, then there is at least one edge among these three vertices in the prime graph of $G$.

Okay, let’s suppose there’s a counterexample $G$, and take one of minimal order. Let $P$ be a Sylow $p$-subgroup of $K$, and $N$ its normaliser in $G$. By the Frattini argument $G=K.N$ and so $G/K \simeq N/(N \cap K)$.

Then there’s a normal series $1 \triangleleft P \triangleleft (N \cap H)=N_H(P) \triangleleft N$, and by Frattini $H=K.(N \cap H)$. But then, $N/(N \cap H)=H.N/H = G/H$ and $(N \cap H)/P$ maps onto $(N \cap H)/(N \cap H \cap K) = H/K$, so this series satisfies the conditions for the three primes $p,q$ and $r$.

But as there is no edge among $p,q$ and $r$ in the prime graph of $G$, there can be no such edge in the prime graph of $N$, and $N$ would be a counterexample of smaller order, unless $N=G$.

Oh, I should have said this before: if there is an edge between two primes in the prime graph of a subgroup (or a quotient) of $G$, then such as edge exists also in the prime group of $G$ (trivial for subgroups, use lifts of elements for quotients).

The only way out is that $N=G$, or that $P$ is a normal subgroup of $G$. Look at quotients $\overline{G}=G/P$ and $\overline{H}=H/P$, take a Sylow $q$-subgroup of $\overline{H}$ and $\overline{M}$ its normaliser in $\overline{G}$.

Frattini again gives $\overline{M}/(\overline{M} \cap \overline{H}) = \overline{G}/\overline{H}$, and $r$ is a prime divisor of the order of $\overline{M}/\overline{Q}$.

Lift the whole schmuck to the lift of $\overline{M}$ in $G$ and get a series of normal subgroups
$1 \triangleleft P \triangleleft Q \triangleleft M$
satisfying the three primes condition, so would give a smaller counter-example unless $M=G$ and $Q$ (the lift of $\overline{Q}$ to $G$) is a normal subgroup of $G$.

Sooner or later, in almost all proofs around the Gruenberg-Kegel result, a Frobenius group enters the picture.

Here, we take an element $x$ in $G$ of order $r$, and consider the subgroup $F$ generated by $Q$ and $x$. The action of $x$ on $Q$ by conjugation is fixed-point free (if not, $G$ would have elements of order $p.r$ or $q.r$ and there is no edge between these prime vertices by assumption).

But then, $F$ is a semi-direct product $Q \rtimes \langle x \rangle$, and again because $G$ has no elements of order $p.r$ nor $q.r$ we have:

• the centraliser-subgroup in $F$ of any non-identity element in $\langle x \rangle$ is contained in $\langle x \rangle$
• the centraliser-subgroup in $F$ of any non-identity element in $Q$ is contained in $Q$

So, $F$ is a Frobenius group with ‘Frobenius kernel’ $Q$. Thompson proved that the Frobenius kernel is a nilpotent group, so a product of its Sylow-subgroups. But then, $Q$ (and therefore $G$) contains an element of order $p.q$, done.

Until now, we’ve looked at actions of groups (such as the $T/I$ or $PLR$-group) or (transformation) monoids (such as Noll’s monoid) on special sets of musical elements, in particular the twelve pitch classes $\mathbb{Z}_{12}$, or the set of all $24$ major and minor chords.

Elephant-lovers recognise such settings as objects in the presheaf topos on the one-object category $\mathbf{M}$ corresponding to the group or monoid. That is, we look at contravariant functors $\mathbf{M} \rightarrow \mathbf{Sets}$.

Last time we’ve encountered the ‘Cube Dance Grap’ which depicts a particular relation among the major, minor, and augmented chords.

Recall that the twelve major chords (numbered for $1$ to $12$) are the ordered triples of tones in $\mathbb{Z}_{12}$ of the form $(n,n+4,n+7)$ (such as the triangle on the left). The twelve minor chords (numbered from $13$ to $24$) are the ordered triples $(n,n+3,n+7)$ (such as the middle triangle). The four augmented chords (numbered from $25$ to $28$) are the triples of the form $(n,n+4,n+8)$ (such as the rightmost triangle).

The Cube Dance Graph relates two of these chords when they share two tones (pitch classes) whereas the remaining tones differ by a halftone.

Picture modified from this post.

We can separate this symmetric binary relation into three sub-relations: the extension of the $P$ and $L$-operations on major and minor chords to the augmented ones (these are transformations), and the remaining relation $U$ which connects the major and minor chords to the augmented chords (and which is not a transformation).

Binary relations on the same set can be composed, so we get a monoid $\mathbf{M}$ generated by the three relations $P,L$ and $U$. The action of $\mathbf{M}$ on the $28$ chords no longer gives us an ordinary presheaf (because $U$ is not a transformation), but a relational presheaf as in the paper On the use of relational presheaves in transformational music theory by Alexandre Popoff.

That is, the action defines a contravariant functor $\mathbf{M} \rightarrow \mathbf{Rel}$ where $\mathbf{Rel}$ is the category (actually a $2$-category) of sets, but with binary relations as morphisms (that is, $Hom(X,Y)$ is all subsets of $X \times Y$), and the natural notion of composition of such relations. The $2$-morphism between relations is that of inclusion.

To compute with monoids generated by binary relations in GAP one needs to download, compile and load the package semigroups, and to represent the binary relations as partitioned binary relations as in the paper by Martin and Mazorchuk.

This is a bit more complicated than working with ordinary transformations:

 P:=PBR([[-13],[-14],[-15],[-16],[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-9],[-10],[-11],[-12],[-25],[-26],[-27],[-28]],[[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[25],[26],[27],[28]]); L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[-10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28]],[[17],[18],[19],[20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5],[6],[7],[8],[25],[26],[27],[28]]); U:=PBR([[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-17,-21,-13,-4,-8,-12],[-5,-1,-9,-18,-14,-22],[-2,-6,-10,-15,-23,-19],[-24,-16,-20,-11,-3,-7]],[[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[25],[25],[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[17,21,13,4,8,12],[5,1,9,18,14,22],[2,6,10,15,23,19],[24,16,20,11,3,7]]); 

But then, GAP quickly tells us that $\mathbf{M}$ is a monoid consisting of $40$ elements.

 gap> M:=Semigroup([P,L,U]); gap> Size(M); 40 

The Semigroups-package can also compute Green’s relations and tells us that there are seven such $R$-classes, four consisting of $6$ elements, two of four, and one of eight elements. These are also visible in the Cayley graph, exactly as last time.

Or, if you prefer the cleaner picture of the Cayley graph from the paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis by Popoff, Andreatta and Ehresmann.

This then allows us to compute the Heyting algebra of the subobject classifier, and all the Grothendieck topologies, at least for the ordinary presheaf topos of $\mathbf{M}$-sets, not for the relational presheaves we need here.

We can consider the same binary relation on the larger set of triads when we add the suspended triads. These are the ordered triples in $\mathbb{Z}_{12}$ of the form $(n,n+5,n+7)$, as in the rightmost triangle below.

There are twelve suspended chords (numbered from $29$ to $40$), so we now have a binary relation $T$ on a set of $40$ triads.

The relation $T$ is too coarse, and the art is to subdivide $T$ is disjoint sub-relations which are musically significant, between major and minor triads, between major/minor and augmented triads, and so on.

For each such partition we can then consider the monoids generated by these sub-relations.

In his paper, Popoff suggest relevant sub-relations $P,L,T_U,T_V$ and $T_U \cup T_V$ of $T$ which in our numbering of the $40$ chords can be represented by these PBR’s (assuming I made no mistakes…ADDED march 24th: I did make a mistake in the definition of L, see comment by Alexandre Popoff, below the corect L):

 P:=PBR([[-13],[-14],[-15],[-16],[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-9],[-10],[-11],[-12],[-25],[-26],[-27],[-28],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35]],[[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[25],[26],[27],[28],[34],[35],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33]]); L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[ -10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28],[-29], [-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40]],[[17], [18], [19], [ 20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5], [6], [7],[8],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40] ]); TU:=PBR([[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[],[],[],[],[],[],[],[],[],[],[],[]],[[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[25],[25],[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[],[],[],[],[],[],[],[],[],[],[],[]]); TV:=PBR([[-29],[-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35],[],[],[],[],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33],[34],[35],[],[],[],[],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]); TUV:=PBR([[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[25,36],[26,37],[27,38],[28,39],[25,40],[25,36],[26,37],[27,38],[28,39],[25,40],[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]); 

The resulting monoids are huge:

 gap> G:=Semigroup([P,L,TU,TV]); gap> Size(G); 473293 gap> H:=Semigroup([P,L,TUV]); gap> Size(H); 994624 

In Popoff’s paper these monoids have sizes respectively $473,293$ and $994,624$. Strangely, the offset is in both cases $144=12^2$. (Added march 24: with the correct L I get the same sizes as in Popoff’s paper).

Perhaps we should try to transform such relational presheaves to ordinary presheaves.

One approach is to use the Grothendieck construction and associate to a set with such a relational monoid action a directed graph, coloured by the elements of the monoid. That is, an object in the presheaf topos of the category
$\xymatrix{C & E \ar[l]^c \ar@/^2ex/[r]^s \ar@/_2ex/[r]_t & V}$
and then we should consider the slice topos over the one-vertex bouquet graph with one loop for each element in the monoid.

If you want to have more details on the musical side of things, for example if you want to know what the opening twelve chords of “Take a Bow” by Muse have to do with the Cube Dance graph, here are some more papers:

A categorical generalization of Klumpenhouwer networks, A. Popoff, M. Andreatta and A. Ehresmann.

From K-nets to PK-nets: a categorical approach, A. Popoff, M. Andreatta and A. Ehresmann.

From a Categorical Point of View: K-Nets as Limit Denotators, G. Mazzola and M. Andreatta.

Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular $12$-gon.

If we move clockwise along the $12$-gon, starting from the endpoint of the longest edge (the root of the chord, here the $0$-vertex) the edges skip $3,2$ and $4$ vertices (for a major chord, here on the left the major $0$-chord) or $2,3$ and $4$ vertices (for a minor chord, here on the right the minor $0$-chord).

The symmetries of the $12$-gon, the dihedral group $D_{12}$, act on the $24$ major- and minor-chords transitively, preserving the type for rotations, and interchanging majors with minors for reflections.

Mathematical Music Theoreticians (MaMuTh-ers for short) call this the $T/I$-group, and view the rotations of the $12$-gon as transpositions $T_k : x \mapsto x+k~\text{mod}~12$, and the reflections as involutions $I_k : x \mapsto -x+k~\text{mod}~12$.

Note that the elements of the $T/I$-group act on the vertices of the $12$-gon, from which the action on the chord-triangles follows.

There is another action on the $24$ major and minor chords, mapping a chord-triangle to its image under a reflection in one of its three sides.

Note that in this case the reflection $I_k$ used will depend on the root of the chord, so this action on the chords does not come from an action on the vertices of the $12$-gon.

There are three such operations: (pictures are taken from Alexandre Popoff’s blog, with the ‘funny names’ removed)

The $P$-operation is reflection in the longest side of the chord-triangle. As the longest side is preserved, $P$ interchanges the major and minor chord with the same root.

The $L$-operation is refection in the shortest side. This operation interchanges a major $k$-chord with a minor $k+4~\text{mod}~12$-chord.

Finally, the $R$-operation is reflection in the middle side. This operation interchanges a major $k$-chord with a minor $k+9~\text{mod}~12$-chord.

From this it is already clear that the group generated by $P$, $L$ and $R$ acts transitively on the $24$ major and minor chords, but what is this $PLR$-group?

If we label the major chords by their root-vertex $1,2,\dots,12$ (GAP doesn’t like zeroes), and the corresponding minor chords $13,14,\dots,24$, then these operations give these permutations on the $24$ chords:

 P:=(1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) L:=(1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16) R:=(1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21) 

Then GAP gives us that the $PLR$-group is again isomorphic to $D_{12}$:

 gap> G:=Group(P,L,R);; gap> Size(G); 24 gap> IsDihedralGroup(G); true 

In fact, if we view both the $T/I$-group and the $PLR$-group as subgroups of the symmetric group $Sym(24)$ via their actions on the $24$ major and minor chords, these groups are each other centralizers! That is, the $T/I$-group and $PLR$-group are dual to each other.

For more on this, there’s a beautiful paper by Alissa Crans, Thomas Fiore and Ramon Satyendra: Musical Actions of Dihedral Groups.

What does this new MaMuTh info learns us more about our Elephant, the Topos of Triads, studied by Thomas Noll?

Last time we’ve seen the eight element triadic monoid $T$ of all affine maps preserving the three tones $\{ 0,4,7 \}$ of the major $0$-chord, computed the subobject classified $\Omega$ of the corresponding topos of presheaves, and determined all its six Grothendieck topologies, among which were these three:

Why did we label these Grothendieck topologies (and corresponding elements of $\Omega$) by $P$, $L$ and $R$?

We’ve seen that the sheafification of the presheaf $\{ 0,4,7 \}$ in the triadic topos under the Grothendieck topology $j_P$ gave us the sheaf $\{ 0,3,4,7 \}$, and these are the tones of the major $0$-chord together with those of the minor $0$-chord, that is the two chords in the $\langle P \rangle$-orbit of the major $0$-chord. The group $\langle P \rangle$ is the cyclic group $C_2$.

For the sheafication with respect to $j_L$ we found the $T$-set $\{ 0,3,4,7,8,11 \}$ which are the tones of the major and minor $0$-,$4$-, and $8$-chords. Again, these are exactly the six chords in the $\langle P,L \rangle$-orbit of the major $0$-chord. The group $\langle P,L \rangle$ is isomorphic to $Sym(3)$.

The $j_R$-topology gave us the $T$-set $\{ 0,1,3,4,6,7,9,10 \}$ which are the tones of the major and minor $0$-,$3$-, $6$-, and $9$-chords, and lo and behold, these are the eight chords in the $\langle P,R \rangle$-orbit of the major $0$-chord. The group $\langle P,R \rangle$ is the dihedral group $D_4$.

More on this can be found in the paper Commuting Groups and the Topos of Triads by Thomas Fiore and Thomas Noll.

The operations $P$, $L$ and $R$ on major and minor chords are reflexions in one side of the chord-triangle, so they preserve two of the three tones. There’s a distinction between the $P$ and $L$ operations and $R$ when it comes to how the third tone changes.

Under $P$ and $L$ the third tone changes by one halftone (because the corresponding sides skip an even number of vertices), whereas under $R$ the third tone changes by two halftones (a full tone), see the pictures above.

The $\langle P,L \rangle = Sym(3)$ subgroup divides the $24$ chords in four orbits of six chords each, three major chords and their corresponding minor chords. These orbits consist of the

• $0$-, $4$-, and $8$-chords (see before)
• $1$-, $5$-, and $9$-chords
• $2$-, $6$-, and $10$-chords
• $3$-, $7$-, and $11$-chords

and we can view each of these orbits as a cycle tracing six of the eight vertices of a cube with one pair of antipodal points removed.

These four ‘almost’ cubes are the NE-, SE-, SW-, and NW-regions of the Cube Dance Graph, from the paper Parsimonious Graphs by Jack Douthett and Peter Steinbach.

To translate the funny names to our numbers, use this dictionary (major chords are given by a capital letter):

The four extra chords (at the N, E, S, and P places) are augmented triads. They correspond to the triads $(0,4,8),~(1,5,9),~(2,6,10)$ and $(3,7,11)$.

That is, two triads are connected by an edge in the Cube Dance graph if they share two tones and differ by an halftone in the third tone.

This graph screams for a group or monoid acting on it. Some of the edges we’ve already identified as the action of $P$ and $L$ on the $24$ major and minor triads. Because the triangle of an augmented triad is equilateral, we see that they are preserved under $P$ and $L$.

But what about the edges connecting the regular triads to the augmented ones? If we view each edge as two directed arrows assigned to the same operation, we cannot do this with a transformation because the operation sends each augmented triad to six regular triads.

Alexandre Popoff, Moreno Andreatta and Andree Ehresmann suggest in their paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis that one might use a monoid generated by relations, and they show that there is such a monoid with $40$ elements acting on the Cube Dance graph.

Popoff claims that usual presheaf toposes, that is contravariant functors to $\mathbf{Sets}$ are not enough to study transformational music theory. He suggest to use instead functors to $\mathbf{Rel}$, that is Sets with as the morphisms binary relations, and their compositions.

Another Elephant enters the room…

(to be continued)