Published in American Mathematical Monthly 105 (1998), no. 4, pp. 359–361.

THE NUMBER OF CONJUGACY CLASSES

Michael Reid

Let G be a finite group, |G| its order, and s the number of its conjugacy classes. Burnside [1, p. 295] uses the theory of representations of finite groups to prove that if |G| is odd, then |G| ≡ s mod 16. He also gives as an exercise [1, p. 320] that if every prime p dividing |G| satisfies p ≡ 1 mod 4, then |G| ≡ s mod 32. Poonen [4] gives an elementary proof (i.e. without using representation theory) of a generalization of the latter statement. Specifically, Poonen shows that if m > 1, and every prime p dividing |G| satisfies p ≡ 1 mod m, then |G| ≡ s mod 2m^{2}.

The purpose of this note is to prove another congruence along the same lines, and to show we have obtained the strongest possible results. We introduce some notation. For m ≥ 1, let G_{m} denote the collection of all finite groups G such that every prime p dividing |G| satisfies p ≡ 1 mod m. Let B(m) denote the greatest common divisor of |G| − s, over all G in G_{m}. Then the above results are reformulated as:

Theorem. (Burnside) B(2) is divisible by 16.

Theorem. (Poonen) If m > 1, then B(m) is divisible by 2m^{2}.

These results are strengthened slightly by the following observation.

Remark. If m > 1, then any prime p ≡ 1 mod m is necessarily odd. Therefore G_{m} ⊆ G_{2}, so B(m) is divisible by B(2), and thus by 16.

If m > 2, we can say slightly more about B(m). In this case, we obtain the desired result under a weaker hypothesis. This result is an easy exercise using representation theory; see for example [5]. Here we give an elementary proof, using Poonen’s technique.

Proposition. If |G| is not divisible by 3, then |G| ≡ s mod 3.

Proof. Poonen [4] shows that the set A = {(x, y) ∈ G^{2} | xy ≠ yx} has cardinality |G|(|G| − s). The set A is clearly in bijection with B = {(x, y, z) ∈ G^{3} | xyz = 1 ≠ zyx} by (x, y) ↦ (x, y, (xy)^{−1}). Now B supports the order 3 permutation (x, y, z) ↦ (y, z, x), which has no fixed points. Therefore |B| = |G|(|G| − s) is divisible by 3, from which the proposition follows. □

Corollary. If m > 2, then B(m) is divisible by 3.

The corollary, along with the theorems of Burnside and Poonen, gives the strongest possible results about B(m). This is the content of the following theorem.

Theorem. If m > 2, then B(m) is the least common multiple of 48 and 2m^{2}. Also B(2) = 16 and B(1) = 1.

Proof. For m > 2, let B′(m) = LCM(48, 2m^{2}). The corollary and the theorems of Burnside and Poonen show that 3, 16 and 2m^{2} each divide B(m). Therefore, their least common multiple, B′(m), divides B(m). For a prime p, let G_{p} denote the non-abelian group

G_{p} = \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) | a, b ∈ ℤ/p^{2}ℤ and a ≡ 1 mod p\}

Typeset by AMS-TEX

of order p^{3}. The center of G_{p} is

\{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) ∈ G_{p} | a ≡ 1 mod p^{2} and b ≡ 0 mod p\} ,

which has order p. The reader can easily verify that the conjugacy class of a non-central element ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) is

\{( \begin{matrix}a′ & b′ \\ 0 & 1 \end{matrix} ) ∈ G_{p} | a′ ≡ a mod p^{2} and b′ ≡ b mod p\} ,

which has size p. Therefore G_{p} has s_{p} = p^{2} + p − 1 conjugacy classes, p of size 1 and p^{2} − 1 of size p. Note also that |G_{p}| − s_{p} = (p + 1)(p − 1)^{2}. For a prime q, let v_{q} denote the q-adic valuation, so v_{q}(n) is the largest integer e such that q^{e} divides n. We will compare v_{q}(B(m)) and v_{q}(B′(m)). Consider several cases.

Case 1. Suppose q ∤ m and q ≠ 2, 3. By Dirichlet’s theorem on primes in arithmetic progressions (and the Chinese remainder theorem), there is a prime p with p ≡ 1 mod m and p ≡ 2 mod q. Then v_{q}(B(m)) ≤ v_{q}(|G_{p}| − s_{p}) = v_{q}((p + 1)(p − 1)^{2}) = 0 = v_{q}(B′(m)). Case 2. Suppose 3 = q ∤ m. There is a prime p with p ≡ 1 mod m and p ≡ 2 mod 9. Then v_{3}(B(m)) ≤ v_{3}((p + 1)(p − 1)^{2}) = 1 = v_{3}(B′(m)). Case 3. Suppose 2 = q ∤ m. There is a prime p with p ≡ 1 mod m and p ≡ 3 mod 8. Then v_{2}(B(m)) ≤ v_{2}((p + 1)(p − 1)^{2}) = 4 = v_{2}(B′(m)). Case 4. Suppose 2 ≠ q|m. Let e = v_{q}(m). There is a prime p with p ≡ 1 mod m and p ≡ 1 + q^{e} mod q^{e+1}. Then v_{q}(B(m)) ≤ v_{q}((p + 1)(p − 1)^{2}) = 2e = v_{q}(B′(m)). Case 5. Suppose 2 = q|m. Let e = v_{2}(m). If e = 1, there is a prime p with p ≡ 1 mod m and p ≡ 3 mod 8. Then v_{2}(B(m)) ≤ v_{2}((p + 1)(p − 1)^{2}) = 4 = v_{2}(B′(m)). If e ≥ 2, there is a prime p with p ≡ 1 mod m and p ≡ 1 + 2^{e} mod 2^{e+1}. Then v_{2}(B(m)) ≤ v_{2}((p + 1)(p − 1)^{2}) = 2e + 1 = v_{2}(B′(m)).

This shows that for every prime q, we have v_{q}(B(m)) ≤ v_{q}(B′(m)). Therefore B(m) = B′(m), so the first statement is proved. For the second statement, note that B(2) divides |G_{3}| − s_{3} = 16, so B(2) = 16. Also, B(1) divides both |G_{3}| − s_{3} = 16 and |G_{2}| − s_{2} = 3, so B(1) = 1. □

Several authors (e.g. [2, 3, 5]) have considered a similar problem with a different type of hypothesis, namely that the order of G is divisible only by some finite set of primes, p_{1}, p_{2}, …, p_{r}. Hirsch [2] also gives an elementary, but more complicated proof of our Proposition. The strongest result under this type of hypothesis appears to be Mann’s theorem [3, p. 83], from which it follows easily that LCM(48, 2m^{2}) divides B(m) for m > 2. It would be interesting to know if Mann’s result is the best possible under this type of hypothesis.

Acknowledgment.

I am grateful to the referee for bringing some of the references to my attention.

References

1. W. Burnside, Theory of Groups of Finite Order, Second Edition, University Press, Cambridge, 1911.

2. K. A. Hirsch, On a theorem of Burnside, The Quarterly Journal of Mathematics, Oxford, Second Series 1 (1950), 97–99. MR 12,156a

3. Avinoam Mann, Conjugacy classes in finite groups, Israel Journal of Mathematics 31 (1978), no. 1, 78–84. MR 58 #22296 4. Bjorn Poonen, Congruences Relating the Order of a Group to the Number of Conjugacy Classes, American Mathematical

Monthly 102 (1995), no. 5, 440–442.

5. R. W. van der Waall, On a theorem of Burnside, Elemente der Mathematik 25 (1970), 136–137.

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