Suppose that we wish to color the vertices of a square with two different colors, say black and
white. We might suspect that there would be24_{= 16}_{different colorings. However, some of}

these colorings are equivalent. If we color the first vertex black and the remaining vertices
white, it is the same as coloring the second vertex black and the remaining ones white since
we could obtain the second coloring simply by rotating the square90*◦* (Figure14.17).

*B* *W*
*W* *W*
*W* *B*
*W* *W*
*W* *W*
*B* *W*
*W* *W*
*W* *B*

14.3. BURNSIDE’S COUNTING THEOREM 163 Burnside’s Counting Theorem offers a method of computing the number of distinguish- able ways in which something can be done. In addition to its geometric applications, the theorem has interesting applications to areas in switching theory and chemistry. The proof of Burnside’s Counting Theorem depends on the following lemma.

**Lemma 14.18.** *Let* *X* *be aG-set and suppose that* *x* *∼y. Then* *Gx* *is isomorphic to* *Gy.*

*In particular,* *|Gx|*=*|Gy|.*

Proof. Let *G* act on *X* by (*g, x*) *7→* *g·x. Since* *x* *∼* *y, there exists a* *g* *∈* *G* such that
*g·x*=*y. Leta∈Gx*. Since

*gag−*1*·y*=*ga·g−*1*y*=*ga·x*=*g·x*=*y,*

we can define a map*ϕ*:*Gx→Gy* by*ϕ*(*a*) =*gag−*1. The map *ϕ*is a homomorphism since

*ϕ*(*ab*) =*gabg−*1=*gag−*1*gbg−*1=*ϕ*(*a*)*ϕ*(*b*)*.*

Suppose that *ϕ*(*a*) = *ϕ*(*b*). Then *gag−*1 =*gbg−*1 or *a*=*b; hence, the map is injective. To*
show that*ϕ*is onto, let *b*be in*Gy*; then*g−*1*bg* is in *Gx* since

*g−*1*bg·x*=*g−*1*b·gx*=*g−*1*b·y*=*g−*1*·y*=*x*;

and *ϕ*(*g−*1*bg*) =*b.*

**Theorem 14.19 Burnside.** *Let* *G* *be a finite group acting on a set* *X* *and let* *k* *denote*
*the number of orbits of* *X. Then*

*k*= 1

*|G|*
∑

*g∈G*

*|Xg|.*

Proof. We look at all the fixed points*x* of all the elements in *g∈G; that is, we look at*
all *g’s and allx’s such thatgx* =*x. If viewed in terms of fixed point sets, the number of*

all *g’s fixing* *x’s is* _{∑}

*g∈G*

*|Xg|.*

However, if viewed in terms of the stabilizer subgroups, this number is
∑
*x∈X*
*|Gx|*;
hence,∑* _{g}_{∈}_{G}_{|}Xg|*=
∑

*x∈X|Gx|*. By Lemma 14.18, ∑

*y∈Ox*

*|Gy|*=

*|Ox| · |Gx|.*

By Theorem14.11 and Lagrange’s Theorem, this expression is equal to* _{|}G|*. Summing over
all of the

*k*distinct orbits, we conclude that

∑
*g∈G*
*|Xg|*=
∑
*x∈X*
*|Gx|*=*k· |G|.*

**Example 14.20.** Let *X* = *{*1*,*2*,*3*,*4*,*5*}* and suppose that *G* is the permutation group
*G*=*{*(1)*,*(13)*,*(13)(25)*,*(25)*}*. The orbits of *X* are* _{{}*1

*,*3

*}*,

*2*

_{{}*,*5

*}*, and

*4*

_{{}*}*. The fixed point sets are

*X*(13)=*{*2*,*4*,*5*}*

*X*(13)(25)=*{*4*}*

*X*_{(25)}=*{*1*,*3*,*4*}.*
Burnside’s Theorem says that

*k*= 1
*|G|*
∑
*g∈G*
*|Xg|*=
1
4(5 + 3 + 1 + 3) = 3*.*
**A Geometric Example**

Before we apply Burnside’s Theorem to switching-theory problems, let us examine the number of ways in which the vertices of a square can be colored black or white. Notice that we can sometimes obtain equivalent colorings by simply applying a rigid motion to the square. For instance, as we have pointed out, if we color one of the vertices black and the remaining three white, it does not matter which vertex was colored black since a rotation will give an equivalent coloring.

The symmetry group of a square, *D*4, is given by the following permutations:

(1) (13) (24) (1432)

(1234) (12)(34) (14)(23) (13)(24)

The group *G* acts on the set of vertices *{*1*,*2*,*3*,*4*}* in the usual manner. We can describe
the different colorings by mappings from *X* into *Y* = *{B, W}* where *B* and *W* represent
the colors black and white, respectively. Each map *f* : *X* *→* *Y* describes a way to color
the corners of the square. Every *σ* *∈D*4 induces a permutation e*σ* of the possible colorings

given by*σ*_{e}(*f*) =*f◦σ* for*f* :*X→Y*. For example, suppose that *f* is defined by
*f*(1) =*B*

*f*(2) =*W*
*f*(3) =*W*
*f*(4) =*W*

and *σ* = (12)(34). Then e*σ*(*f*) = *f* *◦σ* sends vertex 2 to *B* and the remaining vertices to
*W*. The set of all such _{e}*σ* is a permutation group *G*e on the set of possible colorings. Let*X*e
denote the set of all possible colorings; that is, *X*e is the set of all possible maps from*X* to
*Y*. Now we must compute the number of *G-equivalence classes.*e

1. *X*e_{(1)} =*X*e since the identity fixes every possible coloring. * _{|}X*e

*|*= 24 = 16.

2. *X*e(1234) consists of all*f* *∈X*e such that*f* is unchanged by the permutation (1234). In

this case*f*(1) =*f*(2) =*f*(3) =*f*(4), so that all values of*f* must be the same; that is,
either*f*(*x*) =*B* or*f*(*x*) =*W* for every vertex*x* of the square. So * _{|}X*e

_{(1234)}

*|*= 2. 3.

*|X*e(1432)

*|*= 2.

4. For*X*e(13)(24),*f*(1) =*f*(3)and *f*(2) =*f*(4). Thus, *|X*e(13)(24)*|*= 22= 4.

5. * _{|}X*e

_{(12)(34)}

*|*= 4. 6.

*|X*e(14)(23)

*|*= 4.

14.3. BURNSIDE’S COUNTING THEOREM 165
7. For *X*e(13), *f*(1) = *f*(3) and the other corners can be of any color; hence, *|X*e(13)*|* =

23 _{= 8}_{.}

8. * _{|}X*e(24)

*|*= 8.

By Burnside’s Theorem, we can conclude that there are exactly

1 8(2

4_{+ 2}1_{+ 2}2_{+ 2}1_{+ 2}2_{+ 2}2_{+ 2}3_{+ 2}3_{) = 6}

ways to color the vertices of the square.

**Proposition 14.21.** *Let* *G* *be a permutation group of* *X* *and* *X*e *the set of functions from*
*X* *to* *Y. Then there exists a permutation group* *G*e *acting onX, where*e *σ*_{e}*∈G*e *is defined by*
e

*σ*(*f*) =*f◦σ* *for* *σ* *∈G* *and* *f* *∈X. Furthermore, if*e *n* *is the number of cycles in the cycle*
*decomposition of* *σ, then* *|X*e*σ|*=*|Y|n.*

Proof. Let *σ* *∈* *G* and *f* *∈* *X. Clearly,*e *f* *◦σ* is also in *X. Suppose that*e *g* is another
function from *X* to*Y* such thate*σ*(*f*) =*σ*e(*g*). Then for each *x∈X,*

*f*(*σ*(*x*)) =e*σ*(*f*)(*x*) =e*σ*(*g*)(*x*) =*g*(*σ*(*x*))*.*

Since*σ* is a permutation of*X, every elementx′* in*X* is the image of some*x* in*X* under *σ;*
hence, *f* and *g* agree on all elements of *X. Therefore,* *f* =*g* and *σ*e is injective. The map
*σ7→σ*e is onto, since the two sets are the same size.

Suppose that *σ* is a permutation of *X* with cycle decomposition *σ* =*σ*1*σ*2*· · ·σn*. Any

*f* in *X*e*σ* must have the same value on each cycle of *σ. Since there are* *n* cycles and *|Y|*

possible values for each cycle,* _{|}X*e

*σ|*=

*|Y|n*.

**Example 14.22.** Let *X* = *{*1*,*2*, . . . ,*7*}* and suppose that *Y* = *{A, B, C}*. If *g* is the
permutation of *X* given by (13)(245) = (13)(245)(6)(7), then *n* = 4. Any *f* *∈* *X*e*g* must

have the same value on each cycle in *g. There are* * _{|}Y|*= 3 such choices for any value, so

*|X*e

*g|*= 34 = 81.

**Example 14.23.** Suppose that we wish to color the vertices of a square using four different
colors. By Proposition 14.21, we can immediately decide that there are

1 8(4

4_{+ 4}1_{+ 4}2_{+ 4}1_{+ 4}2_{+ 4}2_{+ 4}3_{+ 4}3_{) = 55}

possible ways.

**Switching Functions**

In switching theory we are concerned with the design of electronic circuits with binary
inputs and outputs. The simplest of these circuits is a switching function that has*n*inputs
and a single output (Figure 14.24). Large electronic circuits can often be constructed by
combining smaller modules of this kind. The inherent problem here is that even for a simple
circuit a large number of different switching functions can be constructed. With only four
inputs and a single output, we can construct 65,536 different switching functions. However,
we can often replace one switching function with another merely by permuting the input
leads to the circuit (Figure14.25).

*f* *f*(*x*1*, x*2*, . . . , xn*)
*xn*
*x*2
*x*1
..
.

**Figure 14.24:** A switching function of *n*variables

We define a * switching* or

*of*

**Boolean function***n*variables to be a function from

_{Z}

*n*

2

to _{Z}_{2}. Since any switching function can have two possible values for each binary *n-tuple*
and there are 2*n* _{binary} _{n-tuples,}_{2}2*n* _{switching functions are possible for} _{n}_{variables. In}

general, allowing permutations of the inputs greatly reduces the number of different kinds of modules that are needed to build a large circuit.

*f* *f*(*a, b*)
*a*
*b*
*f* *f*(*b, a*) =*g*(*a, b*)
*a*
*b*

**Figure 14.25:** A switching function of two variables

The possible switching functions with two input variables *a* and *b* are listed in Ta-
ble 14.26. Two switching functions *f* and *g* are equivalent if *g* can be obtained from *f*
by a permutation of the input variables. For example, *g*(*a, b, c*) = *f*(*b, c, a*). In this case
*g* *∼* *f* via the permutation (*acb*). In the case of switching functions of two variables, the
permutation (*ab*) reduces 16 possible switching functions to 12 equivalent functions since

*f*2*∼f*4
*f*3*∼f*5
*f*10*∼f*12
*f*11*∼f*13*.*
Inputs Outputs
*f*0 *f*1 *f*2 *f*3 *f*4 *f*5 *f*6 *f*7
0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 1 1 1 1
1 0 0 0 1 1 0 0 1 1
1 1 0 1 0 1 0 1 0 1
Inputs Outputs
*f*8 *f*9 *f*10 *f*11 *f*12 *f*13 *f*14 *f*15
0 0 1 1 1 1 1 1 1 1
0 1 0 0 0 0 1 1 1 1
1 0 0 0 1 1 0 0 1 1
1 1 0 1 0 1 0 1 0 1

**Table 14.26:** Switching functions in two variables

For three input variables there are 223 = 256 possible switching functions; in the case
of four variables there are 224 = 65*,*536. The number of equivalence classes is too large to

14.3. BURNSIDE’S COUNTING THEOREM 167 reasonably calculate directly. It is necessary to employ Burnside’s Theorem.

Consider a switching function with three possible inputs, *a,* *b, and* *c. As we have*
mentioned, two switching functions *f* and *g* are equivalent if a permutation of the input
variables of*f* gives*g. It is important to notice that a permutation of the switching functions*
is not simply a permutation of the input values *{a, b, c}*. A switching function is a set of
output values for the inputs*a,b, andc, so when we consider equivalent switching functions,*
we are permuting23possible outputs, not just three input values. For example, each binary
triple (*a, b, c*) has a specific output associated with it. The permutation (*acb*) changes
outputs as follows:
(0*,*0*,*0)*7→*(0*,*0*,*0)
(0*,*0*,*1)*7→*(0*,*1*,*0)
(0*,*1*,*0)*7→*(1*,*0*,*0)
..
.
(1*,*1*,*0)*7→*(1*,*0*,*1)
(1*,*1*,*1)*7→*(1*,*1*,*1)*.*

Let*X*be the set of output values for a switching function in*n*variables. Then*|X|*= 2*n*.
We can enumerate these values as follows:

(0*, . . . ,*0*,*1)*7→*0
(0*, . . . ,*1*,*0)*7→*1
(0*, . . . ,*1*,*1)*7→*2
..
.
(1*, . . . ,*1*,*1)*7→*2*n−*1*.*

Now let us consider a circuit with four input variables and a single output. Suppose that we can permute the leads of any circuit according to the following permutation group:

(*a*)*,* (*ac*)*,* (*bd*)*,* (*adcb*)*,*

(*abcd*)*,* (*ab*)(*cd*)*,* (*ad*)(*bc*)*,* (*ac*)(*bd*)*.*

The permutations of the four possible input variables induce the permutations of the output values in Table 14.27.

Hence, there are

1 8(2

16_{+ 2}_{·}_{2}12_{+ 2}_{·}_{2}6_{+ 3}_{·}_{2}10_{) = 9616}

possible switching functions of four variables under this group of permutations. This number will be even smaller if we consider the full symmetric group on four letters.

Group Number Permutation Switching Function Permutation of Cycles

(*a*) (0) 16
(*ac*) (2*,*8)(3*,*9)(6*,*12)(7*,*13) 12
(*bd*) (1*,*4)(3*,*6)(9*,*12)(11*,*14) 12
(*adcb*) (1*,*2*,*4*,*8)(3*,*6*.*12*,*9)(5*,*10)(7*,*14*,*13*,*11) 6
(*abcd*) (1*,*8*,*4*,*2)(3*,*9*,*12*,*6)(5*,*10)(7*,*11*,*13*,*14) 6
(*ab*)(*cd*) (1*,*2)(4*,*8)(5*,*10)(6*,*9)(7*,*11)(13*,*14) 10
(*ad*)(*bc*) (1*,*8)(2*,*4)(3*,*12)(5*,*10)(7*,*14)(11*,*13) 10
(*ac*)(*bd*) (1*,*4)(2*,*8)(3*,*12)(6*,*9)(7*,*13)(11*,*14) 10

**Table 14.27:** Permutations of switching functions in four variables

**Sage** Sage has many commands related to conjugacy, which is a group action. It also has
commands for orbits and stabilizers of permutation groups. In the supplement, we illustrate
the automorphism group of a (combinatorial) graph as another example of a group action
on the vertex set of the graph.

**Historical Note**

William Burnside was born in London in 1852. He attended Cambridge University from
1871 to 1875 and won the Smith’s Prize in his last year. After his graduation he lectured
at Cambridge. He was made a member of the Royal Society in 1893. Burnside wrote
approximately 150 papers on topics in applied mathematics, differential geometry, and
probability, but his most famous contributions were in group theory. Several of Burnside’s
conjectures have stimulated research to this day. One such conjecture was that every group
of odd order is solvable; that is, for a group *G* of odd order, there exists a sequence of
subgroups

*G*=*Hn⊃Hn−*1 *⊃ · · · ⊃H*1 *⊃H*0 =*{e}*

such that*Hi* is normal in*Hi*+1 and *Hi*+1/*Hi* is abelian. This conjecture was finally proven

by W. Feit and J. Thompson in 1963. Burnside’s The Theory of Groups of Finite Order, published in 1897, was one of the first books to treat groups in a modern context as opposed to permutation groups. The second edition, published in 1911, is still a classic.

**14.4**

**Exercises**

**1.** Examples 14.1–14.5 in the first section each describe an action of a group *G* on a
set *X, which will give rise to the equivalence relation defined by* *G-equivalence. For each*
example, compute the equivalence classes of the equivalence relation, the *G-equivalence*
**classes.**

**2.** Compute all*Xg* and all *Gx* for each of the following permutation groups.

(a) *X*=*{*1*,*2*,*3*}*,*G*=*S*3=*{*(1)*,*(12)*,*(13)*,*(23)*,*(123)*,*(132)*}*

(b) *X*=*{*1*,*2*,*3*,*4*,*5*,*6*}*,*G*=*{*(1)*,*(12)*,*(345)*,*(354)*,*(12)(345)*,*(12)(354)*}*

**3.** Compute the *G-equivalence classes ofX* for each of the *G-sets in Exercise*14.4.2. For
each *x∈X* verify that*|G|*=*|Ox| · |Gx|*.

14.4. EXERCISES 169
**4.** Let*G*be the additive group of real numbers. Let the action of *θ∈G*on the real plane

R2 _{be given by rotating the plane counterclockwise about the origin through}_{θ}_{radians. Let}

*P* be a point on the plane other than the origin.
(a) Show that _{R}2 _{is a}_{G-set.}

(b) Describe geometrically the orbit containing*P*.
(c) Find the group*GP*.

**5.** Let*G*=*A*4 and suppose that*G*acts on itself by conjugation; that is,(*g, h*) *7→* *ghg−*1.

(a) Determine the conjugacy classes (orbits) of each element of*G.*
(b) Determine all of the isotropy subgroups for each element of *G.*

**6.** Find the conjugacy classes and the class equation for each of the following groups.

(a) *S*4 (b) *D*5 (c) Z9 (d) *Q*8

**7.** Write the class equation for*S*5 and for *A*5.

**8.** If a square remains fixed in the plane, how many different ways can the corners of the
square be colored if three colors are used?

**9.** How many ways can the vertices of an equilateral triangle be colored using three
different colors?

**10.** Find the number of ways a six-sided die can be constructed if each side is marked
differently with1*, . . . ,*6dots.

**11.** Up to a rotation, how many ways can the faces of a cube be colored with three
different colors?

**12.** Consider12straight wires of equal lengths with their ends soldered together to form
the edges of a cube. Either silver or copper wire can be used for each edge. How many
different ways can the cube be constructed?

**13.** Suppose that we color each of the eight corners of a cube. Using three different
colors, how many ways can the corners be colored up to a rotation of the cube?

**14.** Each of the faces of a regular tetrahedron can be painted either red or white. Up to
a rotation, how many different ways can the tetrahedron be painted?

**15.** Suppose that the vertices of a regular hexagon are to be colored either red or white.
How many ways can this be done up to a symmetry of the hexagon?

**16.** A molecule of benzene is made up of six carbon atoms and six hydrogen atoms,
linked together in a hexagonal shape as in Figure14.28.

(a) How many different compounds can be formed by replacing one or more of the hy- drogen atoms with a chlorine atom?

(b) Find the number of different chemical compounds that can be formed by replacing
three of the six hydrogen atoms in a benzene ring with a*CH*3 radical.

*H*

*H*

*H*
*H*

*H* *H*

**Figure 14.28:** A benzene ring

**17.** How many equivalence classes of switching functions are there if the input variables
*x*1,*x*2, and*x*3 can be permuted by any permutation in*S*3? What if the input variables*x*1,

*x*2,*x*3, and *x*4 can be permuted by any permutation in*S*4?

**18.** How many equivalence classes of switching functions are there if the input variables
*x*1, *x*2, *x*3, and *x*4 can be permuted by any permutation in the subgroup of *S*4 generated

by the permutation (*x*1*x*2*x*3*x*4)?

**19.** A striped necktie has 12 bands of color. Each band can be colored by one of four
possible colors. How many possible different-colored neckties are there?

**20.** A group acts * faithfully* on a

*G-setX*if the identity is the only element of

*G*that leaves every element of

*X*fixed. Show that

*G*acts faithfully on

*X*if and only if no two distinct elements of

*G*have the same action on each element of

*X.*

**21.** Let*p*be prime. Show that the number of different abelian groups of order *pn* (up to
isomorphism) is the same as the number of conjugacy classes in *Sn*.

**22.** Let*a∈G. Show that for anyg∈G,gC*(*a*)*g−*1 =*C*(*gag−*1).

**23.** Let* _{|}G|*=

*pn*be a nonabelian group for

*p*prime. Prove that

*(*

_{|}Z*G*)

*|< pn−*1.

**24.**Let

*G*be a group with order

*pn*

_{where}

_{p}_{is prime and}

_{X}_{a finite}

_{G-set. If}_{X}*G*=*{x∈*

*X*:*gx*=*x* for all *g∈G}* is the set of elements in*X* fixed by the group action, then prove
that* _{|}X| ≡ |XG|* (mod

*p*).

**25.** If *G* is a group of order *pn*, where *p* is prime and *n≥*2, show that*G* must have a
proper subgroup of order*p. Ifn≥*3, is it true that*G*will have a proper subgroup of order
*p*2?