MATHEMATICS

P. M. GUDIVOK

ON REPRESENTATIONS OF FINITE GROUPS OVER QUADRATIC RINGS

(Presented by Academician P. S. Novikov on 19 VI 1964)

Let \(G\) be a finite group, and let \(L\) be a commutative ring with identity. By representations of the group \(G\) over the ring \(L\) we shall mean representations of the group \(G\) by matrices over \(L\). Denote by \(n(G,L)\) the number of indecomposable \(L\)-representations of the group \(G\). The problem is posed:

(A) For which pairs \((G,L)\) is the number \(n(G,L)\) finite?

This problem has been solved for the following rings \(L\): 1) the ring of rational integers \((^{1-5})\); 2) the ring of integers of the field \(R_p(\xi)\) (\(R_p\) is the field of \(p\)-adic numbers, \(\xi^d=1\)) \((^{6-8})\); 3) the ring of residue classes modulo \(m\) \((^9)\).

In addition, the author \((^{6-7})\) and independently Dade \((^8)\) proved the following lemma:

Lemma 1. Let \(F\) be a finite extension of the field of \(p\)-adic numbers \(R_p\), and let \(T\) be the ring of integers of the field \(F\) (with respect to the norm in \(F\)). If a \(p\)-group \(H\) has at least 4 irreducible representations over the field \(F\), then the number \(n(H,T)\) is infinite.

By virtue of Lemma 1, the solution of problem (A) for \(L=T\) is easily reduced to the case where \(G\) is a cyclic \(p\)-group of order \(p^m\) (\(m\leq 2\)); moreover, for \(p>3\) the field \(F\) does not contain a primitive \(p\)-th root of unity. The latter problem is closely connected with the study of modules over the ring \(T[\eta]\) (\(\eta^{p^\alpha}=1;\ 1\leq \alpha \leq 2\)). In the studied cases of problem (A) (for numerical rings of characteristic 0), the ring \(L[\eta]\) was Dedekind, which made it possible to use the well-known theorem of Steinitz \((^{10})\) on modules over Dedekind rings*.

Let \(Q\) be a quadratic extension of the field of rational numbers \(R\); let \(K\) be the ring of algebraic integers of the field \(Q\); let \(Q'\) be a quadratic extension of the field of \(p\)-adic numbers \(R_p\), and let \(K'\) be the ring of integers of the field \(Q'\). In the present note, problem (A) is solved for the rings \(K\) and \(K'\). In this case the rings \(K[\eta]\) and \(K'[\eta]\), generally speaking, will not be Dedekind.

Lemma 2. Let \(H\) be a cyclic \(p\)-group of order \(p^m\) \((1\leq m\leq 2)\), let \(\varepsilon\) be a primitive \(p\)-th root of unity, and let \(T\) be the ring of integers of a finite extension \(F\) of the field \(R_p\). If the group \(H\) has exactly three irreducible \(F\)-representations and the ideal \(I=(1-\varepsilon)\) of the ring \(T[\varepsilon]\) is not principal, then the number \(n(H,T)\) is infinite.

It follows from Lemma 2 that the following hypothesis of Dade \((^8)\) is false: if a \(p\)-group \(H\) has no more than three irreducible \(F\)-representations, then \(n(H,T)<\infty\). (An example is easily constructed.)

Lemma 3 \((^{11})\). Let \(F_1\) and \(F_2\) be finite extensions of the field of \(p\)-adic numbers \(R_p\), with \(F_1\subset F_2\); let \(T_i\) be the ring of integers of the field \(F_i\) \((i=1,2)\), and let \(\Gamma_1\) and \(\Gamma_2\) be two \(T_1\)-representations of a finite group \(G\). The representations \(\Gamma_1\) and \(\Gamma_2\) are \(T_1\)-equivalent if and only if they are \(T_2\)-equivalent.

Next, let \(Q'\) be a quadratic extension of the field of \(p\)-adic numbers \(R_p\), and let \(K'\) be the ring of integers of the field \(Q'\). If \(Q'\) is an unramified extension of the field \(R_p\), then problem (A) for the ring \(K'\) is solved in exactly the same way as for the ring of integers of the \(p\)-adic numbers \((^1)\).

Let now

\[ Q'= \begin{cases} R_p(\sqrt{p}), & \text{if } p\equiv 1 \pmod 4,\\ R_p(\sqrt{p\omega}), & \text{if } p\equiv -1 \pmod 4, \end{cases} \tag{1} \]

* Problem (A) can be solved for any numerical ring \(L\) for which \(L[\eta]\) is a Dedekind ring \((\eta^{p^\alpha}=1;\ 0\leq \alpha \leq 2)\).

($\omega$ is a primitive root of degree $p-1$ of unity). In this case the ring $K'[\varepsilon]$ will be the ring of integers of the field $Q'(\varepsilon)$ ($\varepsilon^p=1$). It is easy to see that the circle-division polynomial $\Phi_p(x)$ of order $p$ decomposes over the field of the form (1) into the product of two irreducible polynomials of the same degree $s=(p-1)/2$: $\Phi_p(x)=f_1(x)f_2(x)$. Hence it follows that the cyclic group $H=(a)$ of order $p$ has exactly 3 irreducible representations over the ring $K'$ of integers of the field $Q'$ of the form (1), namely:

\[ a\to 1,\qquad a\to \widetilde{\varepsilon}_i\quad (i=1,2), \tag{2} \]

where $\widetilde{\varepsilon}_i$ is the matrix corresponding to the operator of multiplication by the primitive root $\varepsilon_i$ of degree $p$ of $1$ in the $K'$-basis $1,\varepsilon_i,\ldots,\varepsilon_i^{s-1}$ of the ring $K'[\varepsilon_i]$ ($f_i(\varepsilon_i)=0;\ i=1,2;\ s=(p-1)/2$).

Any $K'$-representation of the group $H=(a)$ with irreducible components $\widetilde{\varepsilon}_1$ and $\widetilde{\varepsilon}_2$ can be written in the form

\[ a\to \begin{pmatrix} \widetilde{\varepsilon}_1 & \langle\delta\rangle\\ 0 & \widetilde{\varepsilon}_2 \end{pmatrix}, \tag{3} \]

where $\langle\delta\rangle$ is a matrix over $K'$ in which all columns except the last are zero, and the last consists of the coordinates of the element $\delta\in K'[\varepsilon_1]$ in the $K'$-basis $1,\varepsilon_1,\ldots,\varepsilon_1^{s-1}$ of the ring $K'[\varepsilon_1]$.

Theorem 1. Let $Q'$ be a quadratic field of the form (1), $K'$ the ring of integers of the field $Q'$, and $P=(u)$ a prime ideal of the ring $K'[\varepsilon_1]$. All indecomposable $K'$-representations of the cyclic group $H$ of order $p$ are exhausted by the following representations:

\[ \begin{aligned} &1)\quad 1; \qquad 2)\quad \widetilde{\varepsilon}_i\ (i=1,2); \qquad 3)\quad \begin{pmatrix} \widetilde{\varepsilon}_i & \langle 1\rangle\\ 0 & 1 \end{pmatrix}\ (i=1,2);\\[6pt] &4)\quad \begin{pmatrix} \widetilde{\varepsilon}_1 & \langle u^r\rangle\\ 0 & \widetilde{\varepsilon}_2 \end{pmatrix}; \qquad 5)\quad \begin{pmatrix} \widetilde{\varepsilon}_1 & \langle u^r\rangle & \langle 1\rangle\\ 0 & \widetilde{\varepsilon}_2 & 0\\ 0 & 0 & 1 \end{pmatrix} \left(r=0,1,\ldots,\frac{p-3}{2}\right);\\[6pt] &6)\quad \begin{pmatrix} \widetilde{\varepsilon}_1 & \langle u^j\rangle & 0\\ 0 & \widetilde{\varepsilon}_2 & \langle 1\rangle\\ 0 & 0 & 1 \end{pmatrix} \left(j=0,1,\ldots,\frac{p-1}{2}\right);\\[6pt] &7)\quad \begin{pmatrix} \widetilde{\varepsilon}_1 & \langle u^i\rangle & 0 & \langle 1\rangle\\ 0 & \widetilde{\varepsilon}_2 & \langle 1\rangle & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix} \left(i=1,2,\ldots,\frac{p-3}{2};\ p>3\right) \end{aligned} \]

(see the notation (2) and (3)).1

Since the cyclic group $H$ of order $p^2$ has at least 4 irreducible representations over the field $Q'$ of the form (1), it follows, by Lemma 1, that the number $n(H,K')$ in this case is infinite.

Let, further,

\[ Q'= \begin{cases} R_p(\sqrt{p}), & \text{if } p\equiv -1\pmod 4,\\ R_p(\sqrt{p\omega}), & \text{if } p\equiv 1\pmod 4 \end{cases} \tag{4} \]

($\omega$ is a primitive root of degree $p-1$ of $1$); $K'$ is the ring of integers of the field $Q'$; $\varepsilon$ is a primitive root of degree $p$ of $1$; $\pi=(\varepsilon-1)$;

\(t=\sqrt{p}\) or \(\sqrt{p\omega}\); \(I_0=K'[\varepsilon]\), and \(I_r\) is an ideal of the ring \(K'[\varepsilon]\) \((r=1,\ldots,s;\ s=(p-1)/2)\) with \(K'\)-basis \(v_1^{(r)},\ldots,v_{p-1}^{(r)}\): \(v_1^{(r)}=t;\ v_i^{(r)}=\pi^{r-2+i}\) \((i=2,\ldots,d;\ d=p-r);\ v_{d+j}^{(r)}=t\pi^j\) \((j=1,\ldots,r-1)\).

Lemma 4. A group \(H\) of order \(p\) has over the ring \(K'\) of integers of a field \(Q'\) of the form (4) exactly \((p+3)/2\) indecomposable representations, and all of them, except the identity representation, are realized in the ideals \(I_r\) of the ring \(K'[\varepsilon]\) \((r=0,1,\ldots,s)\).

Denote by \(\widetilde T_r\) the matrix corresponding to the operator of multiplication by \(\varepsilon\) in the \(K'\)-basis \(v_1^{(r)},\ldots,v_{p-1}^{(r)}\) of the ideal \(I_r\) \((r=1,\ldots,s;\ s=(p-1)/2)\).

Theorem 2. Let \(K'\) be the ring of integers of a field \(Q'\) of the form (4). A cyclic group of order \(p\) has only the following indecomposable representations over the ring \(K'\):

\[ \begin{gathered} 1)\ 1;\qquad 2)\ \widetilde{\varepsilon};\qquad 3)\ \widetilde T_r\ (r=1,\ldots,s);\qquad 4)\ \begin{pmatrix}\widetilde{\varepsilon}&\langle t^i\rangle\\[2pt]0&1\end{pmatrix}\ (i=0,1); \end{gathered} \]

\[ 5)\ \begin{pmatrix}\widetilde T_r&A\\[2pt]0&1\end{pmatrix}, \quad \text{where }\ A= \begin{pmatrix} 1\\ 0\\ \vdots\\ 0 \end{pmatrix} \quad (r=1,2,\ldots,s;\ s=(p-1)/2); \]

\[ 6)\ \begin{pmatrix}\widetilde T_r&B\\[2pt]0&1\end{pmatrix}, \quad \text{where }\ B= \begin{pmatrix} 0\\ 1\\ 0\\ \vdots\\ 0 \end{pmatrix} \quad (r=1,2,\ldots,s-1;\ p>3); \]

\[ 7)\ \begin{pmatrix}\widetilde T_r&C\\[2pt]0&E\end{pmatrix}, \quad \text{where }\ E=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad C= \begin{pmatrix} E\\ 0\\ \vdots\\ 0 \end{pmatrix} \quad (r=1,\ldots,s-1;\ p>3) \]

(see notations (2) and (3)). The number of representations 1)—7) is equal to \(2p\).

Lemma 4 and Theorem 2 are proved using Lemma 3.

The representations of type 7) show that indecomposable \(K'\)-representations of the abelian \(p\)-group \(H\), even with two irreducible \(Q'\)-components, generally speaking, are not realized in ideals of the group ring \(K'H\) of the group \(H\) over the ring \(K'\). Let us note one more interesting circumstance: if \(S_pH\) is the group ring of an abelian group \(H\) over the ring of integers of the \(p\)-adic numbers \(S_p\), then for two indecomposable \(S_pH\)-modules \(M_1\) and \(M_2\), \(\operatorname{Ext}(M_1,M_2)=0\) if \(M_1\) and \(M_2\) are isomorphic as \(R_pH\)-modules (\(R_p\) is the field of \(p\)-adic numbers). It turns out that this fact does not hold in general for indecomposable \(K'H\)-modules.

It follows from Lemma 2 that the cyclic group of order \(p^2\) possesses infinitely many indecomposable representations over the ring of integers of a field \(Q'\) of the form (4). If \(Q'\) is a ramified quadratic extension of the field of 2-adic numbers \(R_2\), then it also follows from Lemma 2 that the cyclic group of order 4 has infinitely many indecomposable representations over the ring of integers of the field \(Q'\).

Theorem 3. Let \(G\) be a finite group; \(Q'\) a quadratic extension of the field of \(p\)-adic numbers \(R_p\); \(K'\) the ring of integers of the field \(Q'\). The number \(n(G,K')\) is finite if and only if the Sylow \(p\)-subgroup \(H\) of the group \(G\) is cyclic, of order not divisible by \(p^3\), and, moreover, if \((H:1)=p^2\), then \(Q'\) is an unramified extension of the field \(R_p\).

Theorem 4. Let \(G\) be a finite group, \(Q\) a quadratic extension of the field of rational numbers \(R\), and \(K\) the ring of all algebraic integers of the field \(Q\). The number \(n(G,K)\) is finite if and only if, for every prime number \(p\mid(G:1)\), the Sylow \(p\)-subgroup of the group \(G\) is cyclic of order \(p^m\) \((m\leq 2)\), and, if \(m=2\), then the number \(p\) is not ramified in the field \(Q\).

The last theorem is proved using the results of the paper \((^5)\).

Let \(P\) be a prime ideal of the ring \(K\) \((p \equiv 0 \pmod P)\); \(K_P\) the ring of \(P\)-integral numbers of the field \(Q\); \(K'\) the ring of integers of the \(P\)-adic completion of the field \(Q\), and \(Z_p\) the ring of \(p\)-integral rational numbers. It follows from \((12–14)\) that a \(Z_p\)-representation of a \(p\)-group is indecomposable over \(Z_p\) if and only if it is indecomposable over the ring of integers of the \(p\)-adic numbers. It turns out that already for representations of the cyclic group \(H\) of order \(p\) over the ring \(K_P\) this assertion is no longer valid. One can give an example of a representation of the group \(H\) that is indecomposable over \(K_P\) and decomposable over \(K'\).

The author expresses his deep gratitude to S. D. Berman and D. K. Faddeev for a number of valuable suggestions.

Uzhgorod State
University

Received
28 V 1964

REFERENCES

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1. We note that the indecomposable representations of the cyclic group of order $p$ over the ring of integers $K'$ of a field of the form (1) are constructed according to the type of indecomposable representations of the cyclic group of order $p^2$ over the ring of integers of the $p$-adic numbers ([1]). ↩