CONGRUENT ZETA FUNCTIONS. NO.2

YOSHIFUMI TSUCHIMOTO

In this lecture we define and observe some properties of conguent zeta functions.

existence of finite fields II.

Proof of Lemma 1.3 (5). We prove the following more general result

Lemma 2.1. Let K be a field. Let G be a finite subgroup of K^{×}(=multiplicative group of K). Then G is cyclic.

Proof. We first prove the lemma when |G| = ℓ^{k} for some prime number ℓ. In such a case Euler-Lagrange theorem implies that any element g of G has an order ℓ^{s} for some s ∈ ℕ, s ≤ k. Let g_{0} ∈ G be an element which has the largest order m. Then we see that any element of G satisfies the equation

x^{m} = 1.

Since K is a field, there is at most m solutions to the equation. Thus |G| ≤ m. So we conclude that the order m of g_{0} is equal to |G| and that G is generated by g_{0}.

Let us proceed now to the general case. Let us factorize the order |G|.

|G| = ℓ_{1}^{k_{1}}ℓ_{2}^{k_{2}} … ℓ_{t}^{k_{t}} (ℓ_{1}, ℓ_{2}, …, ℓ_{t} : prime number,k_{1}, k_{2}, …, k_{t} ∈ ℤ_{>0}).

Then G may be decomposed into product of p-subgroups

G = G_{1} × G_{2} × ⋯ × G_{t} (|G_{j}| = ℓ_{j}^{k_{j}}(j = 1, 2, 3, …, t)).

By using the first step of this proof we see that each G_{j} is cyclic. Thus we conclude that G is also a cyclic group. □

Exercise 2.1. Let G be a finite abelian group. Assume we have a decomposition |G| = m_{1}m_{2} of the order of G such that m_{1} and m_{2} are coprime. Then show the following:

(1) Let us put

H_{j} = {g ∈ G; g^{m_{j}} = e_{G}} (j = 1, 2)

Then H_{1}, H_{2} are subgroups of G.

(2) |H_{j}| = m_{j} (j = 1, 2).

(3) We have

G = H_{1}H_{2}.

Exercise 2.2. Let G_{1}, G_{2} be finite cyclic groups. Assume |G_{1}| and |G_{2}| are coprime. Show that G_{1} × G_{2} is also cyclic.