Representations of Semidirect Product Groups of 2-by-2 Triangular Matrices with Integer Entries Modulo n

Edgar N. Reyes

Southeastern Louisiana University Hammond, Louisiana, USA 70402

1 Introduction

Let Γ_{2}(n)×_{s}Γ_{2}^{∗}(n) be the semidirect product group of the additive group Γ_{2}(n) of all 2-by-2 upper triangular matrices with entries in ZZ_{n} by the multiplicative group Γ_{2}^{∗}(n) of all invertible matrices in Γ_{2}(n). In this paper, we construct the irreducible representations and determine the number of 1-dimensional representations of Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n).

In our analysis, we will express Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n) as an internal direct product of subgroups Γ_{2}(p_{i}^{e_{i}}) ×_{s} Γ_{2}^{∗}(p_{i}^{e_{i}}) indexed by the prime factor decompostion of n. To study the representation theory, we will use the “little groups” method of Mackey, a standard tool for analyzing the representation theory of semidirect product groups with an abelian normal subgroup. In the course of the discussion, we will describe the irreducible representations of Γ_{2}^{∗}(p_{i}^{e_{i}}), Γ_{2}(p_{i}^{e_{i}}) ×_{s} Γ_{2}^{∗}(p_{i}^{e_{i}}), and Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n).

The group multiplication in Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n) is given by (B_{1}, A_{1})(B_{2}, A_{2}) = (B_{1} + A_{1}B_{2}, A_{1}A_{2}), B_{i} ∈ Γ_{2}(n), A_{i} ∈ Γ_{2}(n)^{∗}, i = 1, 2. This paper is motivated by the representation theory of the semidirect product group ZZ_{n} × ZZ_{n}^{∗} which we discuss in detail in the next section.

2 The Scalars

The group multiplication in G_{n} = ZZ_{n} × ZZ_{n}^{∗} is given by (b_{1}, a_{1})(b_{2}, a_{2}) = (b_{1} + a_{1}b_{2}, a_{1}a_{2}), b_{i} ∈ ZZ_{n}, a_{i} ∈ ZZ_{n}^{∗}, i = 1, 2. Note, ZZ_{n} × {1} is a normal subgroup of G_{n}. To each d ∈ ZZ_{n}, let π^{d} be the irreducible character ZZ_{n} × {1} given by

π_{(b,1)}^{d} = exp (\frac{2πi}{n}db)

where b ∈ ZZ_{n} and exp(w) = e^{w}. For each a ∈ ZZ_{n}^{∗}, let π^{d} · a be the irreducible character of ZZ_{n} × {1} defined by (π^{d} · a)_{(b,1)} = π_{((0,a)(b,1)(0,a)^{−1})}^{d}. Then

π^{d} · a = π^{ad}.

These ZZ_{n}^{∗}-orbits are known and they can be written in the form O(π^{k}) = \{π^{ak} : a ∈ ZZ_{n}^{∗}\} where k|n, k ≥ 1. Moreover, |O(k)| = ϕ (\frac{n}{k}) where ϕ is the Euler totient function. For reference, one can see the discussion following [Theorem 2.2, 1].

Next, we apply the “little groups” method of Mackey [Proposition V.7.1, 2]. Let H_{k} = {a ∈ ZZ_{n}^{∗} : ak = k mod n}. It follows that

|H_{k}| = \frac{ϕ(n)}{ϕ(n/k)}.  (2.1)

To each irreducible character ρ of H_{k}, the induced representation

L^{k,ρ} = Ind_{ZZ_{n}×_{s}H_{k}}^{G_{n}}(π^{k} × ρ)

is irreducible and the dimension of this representation is |O(k)|.

Theorem 1 Let Ĝ_{n} be the set of all irreducible representations of G_{n} (where two representations are identified as the same if they are equivalent representations) and let Ĝ_{n}^{1} be the one-dimensional rperesentations in Ĝ_{n}. Then

(a) |Ĝ_{n}| = ∑_{k|n,k≥1} \frac{ϕ(n)}{ϕ(k)}, and

(b) |Ĝ_{n}^{1}| = \{ \begin{matrix}ϕ(n) & if n is an odd integer \\ 2ϕ(n) & if n is an even integer \end{matrix}

Proof: Since the number of irreducible characters of H_{k} is given by (2.1),

|Ĝ_{n}| = ∑_{k|n,k≥1} |H_{k}| = ∑_{k|n,k≥1} \frac{ϕ(n)}{ϕ(n/k)}.

To prove the second part, note the dimension of L^{k,ρ} is ϕ (\frac{n}{k}). Thus, L^{k,ρ} is a one-dimensional representation if and only if k = n or 2k = n. If n is an odd integer, then the number of one-dimensional characters is |H_{n}| = ϕ(n), see (2.1); while if n is even, the number of one-dimensional characters is |H_{n}| + |H_{(n/2)}| = ϕ(n) + \frac{ϕ(n)}{ϕ(2)}. This completes the proof.

As a consequence, we have obtained an expression for the number (= |Ĝ_{n}|) of conjugacy classes in G_{n}. If G′_{n} is the commutator subgroup of G_{n} and since |G_{n}/G′_{n}| = |Ĝ_{n}^{1}|, then we have a closed-form formula for |G_{n}/G′_{n}|.

3 The Representations of Γ_{2}^{∗}(p^{e})

Here, n = p^{e}, p is a prime number, and e ≥ 1 is an integer. Let N be the normal subgroup of Γ_{2}^{∗}(p^{e}) consisting of matrices of the form ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) where

b ∈ ZZ_{n} and let H consist of matrices of the form ( \begin{matrix}a & 0 \\ 0 & c \end{matrix} ) where a, c ∈ ZZ_{n}^{∗}. It is well-known that Γ_{2}^{∗}(p^{e}) = N ×_{s} H is a semidirect product group.

To each d ∈ ZZ_{n}, let π^{d} be the irreducible character of N given by

π^{d} ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) = exp (\frac{2πi}{n}db) .

Let π^{d} · ( \begin{matrix}a & 0 \\ 0 & c \end{matrix} ) be the character of N defined by

(π^{d} · ( \begin{matrix}a & 0 \\ 0 & c \end{matrix} )) ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) = π^{d} (( \begin{matrix}a & 0 \\ 0 & c \end{matrix} ) ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & c \end{matrix} )^{−1})

A short calculation shows that

π^{d} · ( \begin{matrix}a & 0 \\ 0 & c \end{matrix} ) = π^{ac^{−1}d}

and the H-orbits in N̂ are O(π^{p^{f}}) where 0 ≤ f ≤ e.

Theorem 2 The number of irreducible representations and the number of 1dimensional representations of Γ_{2}^{∗}(p^{e}) are given, respectively, by

(a) |Γ̂_{2}^{∗}(p^{e})| = ∑_{f=0}^{e} \frac{ϕ(p^{e})^{2}}{ϕ(p^{f})} , and

(b) |Γ̂_{2}^{∗}(p^{e})^{1}| = \{ \begin{matrix}ϕ(p^{e})^{2} & if p is an odd prime \\ 2ϕ(2^{e})^{2} & if p = 2. \end{matrix}

Proof: Let H_{p^{f}} = {w ∈ H : π^{p^{f}} · w = π^{p^{f}} }. Since

|H_{p^{f}} | = \frac{|H|}{|O(π^{p^{f}})|} = \frac{ϕ(p^{e})^{2}}{ϕ(p^{e−f})}

and in conjunction with the “little groups” method, we obtain |Γ̂_{2}^{∗}(p^{e})| = ∑_{f=0}^{e} |H_{p^{f}} |. To prove the second assertion, we note that ϕ(p^{e−f}) = 1 iff p is an odd prime and e = f, or p = 2 and e − f ≤ 1

4 The Representations of Γ_{2}^{∗}(n)

Let n = ∏_{i}^{s} p_{i}^{e_{i}} be the prime factor decomposition of n where the e_{i}’s are positive integers and the p_{i}’s are distinct prime numbers. We will consider the subgroups of ZZ_{n}^{∗} given by

H_{i} = {x ∈ ZZ_{n}^{∗} : x = 1 mod ∏_{j≠i}^{s} p_{j}^{e_{j}} } .  (4.2)

Lemma 3 There exists an isomorphism ψ_{i} : ZZ_{(p_{i}^{e_{i}})}^{∗} → H_{i} such that for each i

(a) ψ_{i}(x) = x mod p_{i}^{e_{i}} and

(a) ψ_{i}(x) = 1 mod ∏_{j≠i} p_{j}^{e_{j}}.

Proof: To each x ∈ ZZ_{p_{i}^{e_{i}}}^{∗}, by the Chinese Remainder Theorem there exists a unique y ∈ ZZ_{n}^{∗} satisfying y = x mod p_{i}^{e_{i}} and y = 1 mod ∏_{j≠i} p_{j}^{e_{j}}. Let ψ_{i}(x) = y. Then x ↦ ψ_{i}(x) is a monomorphism from ZZ_{p_{i}^{e_{i}}}^{∗} into H_{i}. Since

|H_{i}| = \frac{ϕ(n)}{ϕ (∏_{j≠i} p_{j}^{e_{j}} )} = ϕ(p_{i}^{e_{i}}) = |ZZ_{p_{i}^{e_{i}}}^{∗}|,

the mapping x ↦ ψ_{i}(x) is surjective as well. This proves the lemma.

A few more mappings need to be defined. First, let ϕ_{i} : ZZ_{p_{i}^{e_{i}}} → ZZ_{n} be the monomorphism given by ϕ_{i}(b) = b ∏_{j≠i} p_{j}^{e_{j}}. Second, let f_{i} : Γ_{2}(p_{i}^{e_{i}}) → Γ_{2}(n) be the monomorphism and f_{i}^{∗} : Γ_{2}^{∗}(p_{i}^{e_{i}}) → Γ_{2}^{∗}(n) the injective mapping defined by

f_{i} ( \begin{matrix}b_{1} & b_{2} \\ 0 & b_{3} \end{matrix} ) = ( \begin{matrix}ϕ_{i}(b_{1}) & ϕ_{i}(b_{2}) \\ 0 & ϕ_{i}(b_{3}) \end{matrix} )

and

f_{i}^{∗} ( \begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) = ( \begin{matrix}ψ_{i}(a_{1}) & ϕ_{i}(a_{2}) \\ 0 & ψ_{i}(a_{3}) \end{matrix} ) .

Since ZZ_{n} = ∑_{k=1}^{s} ϕ_{i} (ZZ_{p_{k}^{e_{k}}} ) is an internal direct sum, the following corollary is an immediate consequence.

Corollary 4 Γ_{2}(n) = ∑_{k=1}^{s} f_{k}(Γ_{2}(p_{k}^{e_{k}})) is an internal direct sum.

For brevity, we omit the direct calculations involved in the proof of the next two corollaries.

Corollary 5 Let i ≠ j, a ∈ ZZ_{p_{i}^{e_{i}}}^{∗}, b ∈ ZZ_{p_{i}^{e_{i}}}, and c ∈ ZZ_{p_{j}^{e_{j}}}^{∗}. Then

(a) ϕ_{i}(ab) = ψ_{i}(a)ϕ_{i}(b),

(b) ψ_{j}(c)ϕ_{i}(b) = ϕ_{i}(b), and

(c) ϕ_{i}(a)ϕ_{i}(b) = ϕ_{i}(qab) where q = ∏_{k≠i} p_{k}^{e_{k}} mod p_{i}^{e_{i}}.

Corollary 6 f_{i}^{∗} is a monomorphism.

Theorem 7 Γ_{2}^{∗}(n) = ∏_{k=1}^{s} f_{k}^{∗}(Γ_{2}^{∗}(p_{k}^{e_{k}})) is an internal direct product.

Proof: Given A ∈ Γ_{2}^{∗}(n), we will show for each k there exists a unique X^{k} ∈ Γ_{2}^{∗}(p_{k}^{e_{k}}) satisfying A = ∏_{k=1}^{s} f_{k}^{∗}(X^{k}). Let a_{i,j} be the (i, j)-entry of A. Note, ZZ_{n}^{∗} = ∏_{i=1}^{s} H_{i} is a well-known internal direct product which follows from the Chinese Remainder Theorem. Since a_{i,i} ∈ ZZ_{n}^{∗}, consequently we obtain

a_{i,i} = ∏_{k=1}^{s} ψ_{k}(x_{i,i}^{k})  (4.3)

for some x_{i,i}^{k} ∈ ZZ_{p_{k}^{e_{k}}}^{∗}, cf. Lemma 3. Due to the remark before Corollary 4, we obtain

a_{1,2} = ∑_{k=1}^{s} ϕ_{k}(x_{1,2}^{k})  (4.4)

for some x_{1,2}^{k} ∈ ZZ_{p_{k}^{e_{k}}}.

Let X^{k} be the invertible upper triangular matrix whose (i, j)-entry is x_{i,j}^{k} where x_{2,1}^{k} = 0. Then the diagonal (i, i)-entry of ∏_{k=1}^{s} f_{k}^{∗}(X^{k}) is a_{i,i} by (4.3).

Since ϕ_{k_{1}}(x_{i,j}^{k_{1}})ϕ_{k_{2}}(x_{i,j}^{k_{2}}) = 0 if k_{1} ≠ k_{2}, the (1, 2)-entry of ∏_{k=1}^{s} f_{k}^{∗}(X^{k}) is

∑_{k=1}^{s} ψ_{1}(x_{1,1}^{1})ψ_{2}(x_{1,1}^{2})...ψ_{k−1}(x_{1,1}^{k−1})ϕ_{k}(x_{1,2}^{k})ψ_{k+1}(x_{2,2}^{k+1})...ψ_{s}(x_{2,2}^{s})

which by Corollary 5 reduces to ∑_{k=1}^{s} ϕ_{k}(x_{1,2}^{k}) (= a_{i,j}). Hence, A = ∏_{k=1}^{s} f_{k}^{∗}(X^{k}). The uniqueness of the X^{k}’s follows from (4.3) and (4.4). This completes the proof.

Since the irreducible representations of a direct product of groups are obtained by taking the product of the irreducible representations of the individual groups, a direct consequence of Theorem 7 and Theorem 2 is the following.

Corollary 8 The number of irreducible representations and number of 1-dimensional representations of Γ_{2}^{∗}(n) are given, respectively, by

(a) |Γ̂_{2}^{∗}(n)| = ∏_{i=0}^{s} ∑_{f=0}^{e_{i}} \frac{ϕ(p_{i}^{e_{i}})^{2}}{ϕ(p_{i}^{f})} , and

(b) |Γ̂_{2}^{∗}(n)^{1}| = \{ \begin{matrix}ϕ(n)^{2} & if n is an odd integer \\ 2ϕ(n)^{2} & if n is an even integer \end{matrix}

5 Main Section

Let n ≥ 2 be an arbitrary integer and let M̂_{n}^{1} be the set of 1-dimensional representations of Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n). We will determine the cardinality |M̂_{n}^{1}| of M̂_{n}^{1} and we will construct the irreducible representations of Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n).

As we have denoted earlier, n = ∏_{i}^{s} p_{i}^{e_{i}} is the prime factor decomposition of n where the e_{i}’s are positive integers and the p_{i}’s are distinct prime numbers. Let Im(f_{i}) and Im(f_{i}^{∗}) be the range of f_{i} and of f_{i}^{∗}, respectively.

Theorem 9 The following is an internal direct product of subgroups

Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n) = ∏_{i=1}^{s} f_{i} (Γ_{2}(p_{i}^{e_{i}})) ×_{s} f_{i}^{∗} (Γ_{2}^{∗}(p_{i}^{e_{i}})) .

Proof: Let (B, A) ∈ Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n). By Corollary 4,

B = ∑_{i=1}^{s} B_{i}

for some B_{i} ∈ Im(f_{i}). Likewise, by Theorem 7 there exist A_{i} ∈ Im(f_{i}^{∗}) such that

A = ∏_{i=1}^{s} A_{i}

for some A_{i} ∈ Im(f_{i}^{∗}).

Let

A_{k} = ( \begin{matrix}ψ_{k}(a_{1}) & ϕ_{k}(a_{2}) \\ 0 & ψ_{k}(a_{3}) \end{matrix} ) and B_{j} = ( \begin{matrix}ϕ_{j}(b_{1}) & ϕ_{j}(b_{2}) \\ 0 & ϕ_{j}(b_{3}) \end{matrix} ) .

First, it can be verified directly that f_{i} (Γ_{2}(p_{i}^{e_{i}})) ×_{s} f_{i}^{∗} (Γ_{2}^{∗}(p_{i}^{e_{i}})) is a subgroup of Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n) for each i. Furthermore, for any k ≠ j, by using Corollary 5 we find that A_{k}B_{j} = B_{j}. Then inductively we obtain

(B, A) = (∑_{i=1}^{s} B_{i}, ∏_{i=1}^{s} A_{i})
= (B_{1} + ... + B_{s−1}, A_{1}...A_{s−1})(B_{s}, A_{s})
= ∏_{i=1}^{s}(B_{s}, A_{s}).  (5.5)

The uniqueness of the product in (5.5) is assured by Corollary 4 and Theorem 9. This completes the proof of the Theorem.

The next theorem is a result that one might have anticipated.

Theorem 10 The following is an isomorphism of groups

Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n) ≃ ∏_{i=1}^{s} Γ_{2}(p_{i}^{e_{i}}) ×_{s} Γ_{2}^{∗}(p_{i}^{e_{i}}).

Proof: By Theorem 9, it suffices to show that Γ_{2}(p_{i}^{e_{i}}) ×_{s} Γ_{2}^{∗}(p_{i}^{e_{i}}) is isomorphic to f_{i} (Γ_{2}(p_{i}^{e_{i}})) ×_{s} f_{i}^{∗} (Γ_{2}^{∗}(p_{i}^{e_{i}})). Let (B, A) ∈ Γ_{2}(p_{i}^{e_{i}}) ×_{s} Γ_{2}^{∗}(p_{i}^{e_{i}}) and suppose

A = ( \begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) and B = ( \begin{matrix}b_{1} & b_{2} \\ 0 & b_{3} \end{matrix} ) .

Consider the mapping f defined by

f(B, A) = (( \begin{matrix}ϕ_{i}(b_{1}) & ϕ_{i}(qb_{2}) \\ 0 & ϕ_{i}(b_{3}) \end{matrix} ) , f_{i}^{∗}(A))

where q is defined in Corollary 5. Clearly, f(B, A) ∈ f_{i} (Γ_{2}(p_{i}^{e_{i}}))×_{s}f_{i}^{∗} (Γ_{2}^{∗}(p_{i}^{e_{i}})). Let

A′ = ( \begin{matrix}a′_{1} & a′_{2} \\ 0 & a′_{3} \end{matrix} ) and B′ = ( \begin{matrix}b′_{1} & b′_{2} \\ 0 & b′_{3} \end{matrix} ) .

We consider the following calculations:

f((B, A)(B′, A′)) = f(B + AB′, AA′)
= f (B + ( \begin{matrix}a_{1}b′_{1} & a_{1}b′_{2} + a_{2}b′_{3} \\ 0 & a_{3}b′_{3} \end{matrix} ) , AA′)
= f (( \begin{matrix}b_{1} + a_{1}b′_{1} & b_{2} + a_{1}b′_{2} + a_{2}b′_{3} \\ 0 & b_{3} + a_{3}b′_{3} \end{matrix} ) , AA′)
= (( \begin{matrix}ϕ_{i}(b_{1} + a_{1}b′_{1}) & ϕ_{i}(q(b_{2} + a_{1}b′_{2} + a_{2}b′_{3})) \\ 0 & ϕ_{i}(b_{3} + a_{3}b′_{3}) \end{matrix} ) , f_{i}^{∗}(AA′))

= (( \begin{matrix}ϕ_{i}(b_{1}) & ϕ_{i}(qb_{2}) \\ 0 & ϕ_{i}(b_{3}) \end{matrix} ) + ( \begin{matrix}ψ_{i}(a_{1})ϕ_{i}(b′_{1}) & ψ_{i}(a_{1})ϕ_{i}(qb′_{2}) + ϕ_{i}(a_{2})ϕ_{i}(b′_{3}) \\ 0 & ψ_{i}(a_{3})ϕ_{i}(b′_{3}) \end{matrix} ) ,
f_{i}^{∗}(AA′))
= (( \begin{matrix}ϕ_{i}(b_{1}) & ϕ_{i}(qb_{2}) \\ 0 & ϕ_{i}(b_{3}) \end{matrix} ) , f_{i}^{∗}(A)) (( \begin{matrix}ϕ_{i}(b′_{1}) & ϕ_{i}(qb′_{2}) \\ 0 & ϕ_{i}(b′_{3}) \end{matrix} ) , f_{i}^{∗}(A′))
= f(B, A)f(B′, A′)

Hence, f is a homomorphism. Since the mapping b_{2} ↦ qb_{2} is an automorphism of ZZ_{p_{i}^{e_{i}}}, f is an isomorphism. This completes the proof.

To analyze the representation theory of Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n), it will be sufficient to analyze the representation theory of Γ_{2}(p^{e}) ×_{s} Γ_{2}^{∗}(p^{e}) where p is a prime integer and e is a positive integer.

In the following, we let I_{2} and 0_{2} be the 2-by-2 identity matrix and the 2by-2 matrix consisting entirely of 0’s, respectively. To each D = ( \begin{matrix}d_{1} & d_{2} \\ 0 & d_{3} \end{matrix} ) ∈ Γ_{2}(n), let π^{D} be the character of the subgroup Γ_{2}(n) × {I_{2}} of M_{n} defined by

π^{D}(B, I_{2}) = exp (\frac{2πi}{n}(b_{1}d_{1} + b_{2}d_{2} + b_{3}d_{3}))

where B = ( \begin{matrix}b_{1} & b_{2} \\ 0 & b_{3} \end{matrix} ) ∈ Γ_{2}(n). Given A = ( \begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) ∈ Γ_{2}^{∗}(n), let (π^{D})·A be

the character of Γ_{2}(n) defined by (π^{D} ·A)(B, I_{2}) = π^{D}((0_{2}, A)(B, I_{2})(0_{2}, A)^{−1}). Since (0_{2}, A)(B, I_{2})(0_{2}, A)^{−1} = (AB, A)(0_{2}, A^{−1}) = (AB, I_{2}) and

AB = ( \begin{matrix}a_{1}b_{1} & a_{1}b_{2} + a_{2}b_{3} \\ 0 & a_{3}b_{3} \end{matrix} ) ,

we see that

(π^{D} · A)(B, I_{2}) = exp (\frac{2πi}{n} (d_{1}(a_{1}b_{1}) + d_{2}(a_{1}b_{2} + a_{2}b_{3}) + d_{3}(a_{3}b_{3})))
= exp (\frac{2πi}{n} (b_{1}(a_{1}d_{1}) + b_{2}(a_{1}d_{2}) + b_{3}(a_{2}d_{2} + a_{3}d_{3}))) .

Thus,

(π^{D}) · A = π^{D∗A}

where

D ∗ A = ( \begin{matrix}a_{1}d_{1} & a_{1}d_{2} \\ 0 & a_{2}d_{2} + a_{3}d_{3} \end{matrix} ) .

As can be expected, D ∗ (A_{1}A_{2}) = (D ∗ A_{1}) ∗ A_{2} for A_{1}, A_{2} ∈ Γ_{m}^{∗}(n). We will determine the structure of the orbits of π^{D} for various D′s.

Lemma 11 Let n = p^{e} and let k_{1}, k_{2} be nonnegative integers. Suppose ⟨p^{k_{2}}⟩ is the subgroup of ZZ_{n} generated by p^{k_{2}} and O(p^{k_{3}}) = {ap^{k_{3}} : a ∈ ZZ_{n}^{∗}}. Then

O(p^{k_{3}}) + ⟨p^{k_{2}}⟩ = \{ \begin{matrix}⟨p^{k_{2}}⟩ if k_{3} ≥ k_{2} \\ O(p^{k_{3}}) if k_{3} < k_{2} \end{matrix}

Proof: If k_{3} ≥ k_{2}, then ap^{k_{3}} ∈ ⟨p^{k_{2}}⟩ and O(p^{k_{3}}) ⊆ ⟨p^{k_{2}}⟩. To prove the second part, we note O(p^{k_{3}}) = {x ∈ ZZ_{p^{e}} : gcd(x, p^{e}) = p^{k_{3}}} where gcd(,) represents the greatest common divisor [Theorem 2.2, 1]. If k_{3} < k_{2}, then for any b ∈ ZZ_{p^{e}} we see that

gcd(ap^{k_{3}} + bp^{k_{2}}, p^{e}) = p^{k_{3}} · gcd(a + bp^{k_{2}−k_{3}}, p^{e−k_{3}})
= p^{k_{3}} · 1
= p^{k_{3}}.

Thus, ap^{k_{3}} + bp^{k_{2}} ∈ O(p^{k_{3}}) and O(p^{k_{3}}) + ⟨p^{k_{2}}⟩ = O(p^{k_{3}}) when k_{3} < k_{2}. This proves the lemma.

Lemma 12 Let n = p^{e} and 1 ≤ k ≤ e. Then

O(1) ∪ ... ∪ O(p^{k−1}) ∪ ⟨p^{k}⟩ = ZZ_{n}.  (5.6)

Proof: Note, the left side of (5.6) is a disjoint union of sets. To prove the lemma, it will be sufficient to show that the sum of the cardinalities of the sets in the disjoint union is p^{e}. Note, |⟨p^{k}⟩| = p^{e−k} and |O(p^{f})| = ϕ(p^{e−f}). Since p^{f} + ϕ(p^{f+1}) = p^{f+1}, inductively we obtain

= ϕ(p^{e}) + ϕ(p^{e−1}) + ... + ϕ(p^{e−k+1}) + p^{e−k}
= ϕ(p^{e}) + ϕ(p^{e−1}) + ... + p^{e−k+1}
⋮
= ϕ(p^{e}) + p^{e−1}
= p^{e}

This completes the proof.

For n = p^{e} and 0 ≤ k_{1}, k_{2} ≤ e, let S_{k_{1}} = {a ∈ ZZ_{n}^{∗} : ap^{k_{1}} = p^{k_{1}} mod n} and let R_{k_{1}} be a complete set of representatives of the coset space ZZ_{n}^{∗}/S_{k_{1}}. For k_{1} ≥ k_{2}, a ∈ R_{k_{1}}, and k_{2} ≥ k_{3}, we let

α(a, k_{1}, k_{2}, k_{3}) = O ( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} ) .

While if k_{2} > k_{1}, a ∈ R_{k_{2}}, and k_{2} ≥ k_{3}, we let

β(a, k_{1}, k_{2}, k_{3}) = O ( \begin{matrix}p^{k_{1}} & ap^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} ) .

Lemma 13 Let n = p^{e}. If α(a, k_{1}, k_{2}, k_{3}) = α(a′, k′_{1}, k′_{2}, k′_{3}) or β(a, k_{1}, k_{2}, k_{3}) = β(a′, k′_{1}, k′_{2}, k′_{3}), then a = a′, k_{1} = k′_{1}, k_{2} = k′_{2}, and k_{3} = k′_{3}.

Proof: If α(a, k_{1}, k_{2}, k_{3}) = α(a′, k′_{1}, k′_{2}, k′_{3}), then there exists ( \begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) ∈

Γ_{2}^{∗}(n) such that

( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} ) ∗ ( \begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) = ( \begin{matrix}a′p^{k′_{1}} & p^{k′_{2}} \\ 0 & p^{k′_{3}} \end{matrix} )
( \begin{matrix}a_{1}ap^{k_{1}} & a_{1}p^{k_{2}} \\ 0 & a_{2}p^{k_{2}} + a_{3}p^{k_{3}} \end{matrix} ) =

Since O(p^{k}) = O(p^{k′}) iff k = k′, from the (1, 1)- and (1, 2)-entries of the previous matrix calculation we see that k_{1} = k′_{1} and k_{2} = k′_{2}. Invoking Lemma 11, if k_{3} = k_{2}, then p^{k′_{3}} = a_{2}p^{k_{2}} + a_{3}p^{k_{3}} ∈ ⟨p^{k_{2}}⟩ and k_{3} = k_{2} = k′_{2} ≥ k′_{3} ≥ k_{2} ≥ k_{3}; while if k_{3} < k_{2}, then p^{k′_{3}} = a_{2}p^{k_{2}} + a_{3}p^{k_{3}} ∈ O(p^{k_{3}}). In any case, k_{3} = k′_{3}. For the present case, we have k_{1} ≥ k_{2} and since a_{1}p^{k_{2}} = p^{k_{2}}, we see that a_{1} ∈ S_{k_{2}} ⊆ S_{k_{1}}. Consequently, a_{1}p^{k_{1}} = p^{k_{1}} and since a_{1}ap^{k_{1}} = a′p^{k_{1}}, we have ap^{k_{1}} = a′p^{k_{1}}. Thus, a = a′. The other case can be proved similarly. This proves the lemma.

Lemma 14 Let n = p^{e} and A = ( \begin{matrix}x & y \\ 0 & z \end{matrix} ) ∈ Γ_{2}(n). Then A ∈ α(a, k_{1}, k_{2}, k_{3})

or A ∈ β(a, k_{1}, k_{2}, k_{3}) for some a, k_{1}, k_{2}, k_{3}.

Proof: Choose 0 ≤ k_{1}, k_{2} ≤ e, a_{x}, a_{y} ∈ ZZ_{n}^{∗} such that x = a_{x}p^{k_{1}} and y = a_{y}p^{k_{2}}. We will consider two cases. First, suppose k_{1} ≥ k_{2}. Then

( \begin{matrix}x & y \\ 0 & z \end{matrix} ) = ( \begin{matrix}a_{x}p^{k_{1}} & a_{y}p^{k_{2}} \\ 0 & z \end{matrix} ) ∈ O ( \begin{matrix}(a_{y})^{−1}a_{x}p^{k_{1}} & p^{k_{2}} \\ 0 & z \end{matrix} ) .

Since (a_{y})^{−1}a_{x}S_{k_{1}} = aS_{k_{1}} for some a ∈ R_{k_{1}}, we get

( \begin{matrix}x & y \\ 0 & z \end{matrix} ) ∈ O ( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & z \end{matrix} ) .

But by Lemma 12, z ∈ ⟨p^{k_{2}}⟩ or z ∈ O(p^{f}) for some f < k_{2}. Note,

( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & p^{k_{2}} \end{matrix} ) ∗ ( \begin{matrix}1 & (b − 1) \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & bp^{k_{2}} \end{matrix} )

and

( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & p^{f} \end{matrix} ) ∗ ( \begin{matrix}1 & 0 \\ 0 & a \end{matrix} ) = ( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & ap^{f} \end{matrix} )

Thus, if k_{1} ≥ k_{2} then A ∈ α(a, k_{1}, k_{2}, k_{3}) for some a, k_{1}, k_{2}, k_{3}. For the second case, when k_{2} > k_{1}, it can be shown similarly that A ∈ β(a, k_{1}, k_{2}, k_{3}). This completes the proof.

Now, we apply the “little groups” method to the group Γ_{2}(p^{e}) ×_{s} Γ_{2}^{∗}(p^{e}). To each α(a, k_{1}, k_{2}, k_{3}) and β(a, k_{1}, k_{2}, k_{3}), we associate the subgroups

H_{α(a,k_{1},k_{2},k_{3})} = \{A ∈ Γ_{2}^{∗}(p^{e}) : ( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} ) ∗ A = ( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} )\}
= \{( \begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) ∈ Γ_{2}^{∗}(p^{e}) : a_{1}p^{k_{2}} = p^{k_{2}}, a_{2}p^{k_{2}} + a_{3}p^{k_{3}} = p^{k_{3}}\}

and

H_{β(a,k_{1},k_{2},k_{3})} = \{A ∈ Γ_{2}^{∗}(p^{e}) : ( \begin{matrix}p^{k_{1}} & ap^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} ) ∗ A = ( \begin{matrix}p^{k_{1}} & ap^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} )\}
= \{( \begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) ∈ Γ_{2}^{∗}(p^{e}) : a_{1}p^{k_{1}} = p^{k_{1}}, a_{2}ap^{k_{2}} + a_{3}p^{k_{3}} = p^{k_{3}}\} .

Also, we associate the matrices

D_{α(a,k_{1},k_{2},k_{3})} = ( \begin{matrix}ap^{k_{1}} & p^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} ) and D_{β(a,k_{1},k_{2},k_{3})} = ( \begin{matrix}p^{k_{1}} & ap^{k_{2}} \\ 0 & p^{k_{3}} \end{matrix} ) .

Recall, α(a, k_{1}, k_{2}, k_{3}) where a ∈ R_{k_{1}} is defined for k_{1} ≥ k_{2}, and β(a, k_{1}, k_{2}, k_{3}) where a ∈ R_{k_{2}} is defined for k_{2} > k_{1}. Now, we can describe the representation theory of Γ_{2}(p^{e}) ×_{s} Γ_{2}^{∗}(^{e}).

Theorem 15 Let n = p^{e} and let σ ∈ {α, β}. Suppose H = H_{σ(a,k_{1},k_{2},k_{3})}, D = D_{σ(a,k_{1},k_{2},k_{3})}, and ρ is an irreducible representation of H. Then

Ind_{Γ_{2}(n)×_{s}H}^{Γ_{2}(n)×_{s}Γ_{2}^{∗}(n)}(π^{D} × ρ)

is an irreducible representation whose dimension is dim(ρ) · |Γ_{2}^{∗}(n)/H| where dim(ρ) is the dimension of ρ. Moreover, each irreducible representation (up to equivalence) of Γ_{2}(n) ×_{s} Γ_{2}^{∗}(n) can be induced in this way.

Proof: The various orbits α(a, k_{1}, k_{2}, k_{3}) and β(a, k_{1}, k_{2}, k_{3}) are disjoint and their union is Γ_{2}(n), c.f. Lemmas 13 and 14. The proof of this theorem now follows from the “little groups” method.

Theorem 16 Γ_{2}(n)×_{s}Γ_{2}^{∗}(n) and Γ_{2}^{∗}(n) have the same number of 1-dimensional representations, i.e., |M̂_{n}^{1}| = |Γ̂_{2}^{∗}(n)^{1}|

Proof: First, consider the case when n = p^{e}. Using the notation from Theorem 15, the irreducible representation

Ind_{Γ_{2}(n)×_{s}H}^{Γ_{2}(n)×_{s}Γ_{2}^{∗}(n)}(π^{D} × ρ)

is one-dimensional iff dim(ρ) = 1 and H = Γ_{2}^{∗}(n). Thus, the number of 1-dimensional representations of M_{n} is |Γ̂_{2}^{∗}(n)^{1}|.

Now, let n be an arbitrary integer where n = ∏_{i}^{s} p_{i}^{e_{i}} is the prime factor decomposition of n. By Theorem 10 and 7, it follows that

M̂_{n}^{1} = ∏_{i=1}^{s} |Γ̂_{2}^{∗}(p_{i}^{e_{i}})^{1}| = |Γ̂_{2}^{∗}(n)^{1}|.

This completes the proof.

REFERENCES

1. T.M. Apostol , Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, 1976.

2. B. Simon , Representations of Finite and Compact Groups, Graduate Texts in Mathematics, Vol. 10, American Mathematical Society 1991.