MATHEMATICS

S. D. BERMAN

ON THE THEORY OF INTEGRAL REPRESENTATIONS OF FINITE GROUPS

(Presented by Academician L. S. Pontryagin on 26 II 1964)

I. Let \(G=\langle a\rangle\) be a cyclic group of order \(mp^\alpha\) \(((m,p)=1;\ p\) prime). In \((^1)\) (see also \((^2)\)) it is shown that if \(\alpha \geqslant 3\), then the group \(G\) has indecomposable representations of arbitrarily large degree over the rings of rational integers \(Z\), \(p\)-adic integers \(J_p\), and \(p\)-integral rational numbers \(R_p\). If \(\alpha=1\), then, as established in \((^3)\), all indecomposable \(Z\)-representations (\(R_p\)-representations) of the group \(G\) are realized in ideals of the group ring \(ZG\) \((R_pG)\).

Let \(d_1,\ldots,d_s\) be all the distinct positive divisors of the number \(mp^2\), and let \(\varepsilon_i\) be a primitive \(d_i\)-th root of unity \((i=1,\ldots,s)\). The irreducible \(R_p\)-representations of the group \(G=\langle a\rangle\) of order \(mp^2\) are exhausted by the representations

\[ \Gamma_1,\ldots,\Gamma_s,\quad \text{where } \Gamma_i:\ a \to \widetilde{\varepsilon}_i, \]

and \(\widetilde{\varepsilon}_i\) is the operator of multiplication by \(\varepsilon_i\) in the ring \(M_i=R_p[\varepsilon_i]\) \((i=1,\ldots,s)\).

If

\[ \frac{d_i}{d_j}\ne p^\gamma \quad (\gamma \text{ an integer}), \]

then for the \(G\)—\(R_p\)-modules \(M_i\) and \(M_j\) the condition

\[ \operatorname{Ext}^1(M_i,M_j)=0 \]

holds. Therefore the indecomposable \(R_p\)-representations of the group \(G\) can contain only irreducible diagonal components of the form:

\[ \Gamma^{(1)}=\widetilde{\eta\xi};\qquad \Gamma^{(2)}=\widetilde{\eta\varepsilon};\qquad \Gamma^{(3)}=\widetilde{\eta}, \]

where \(\eta\) is an arbitrary but fixed root of unity of degree \(t\mid m\); \(\xi\) and \(\varepsilon\) are, respectively, primitive roots of unity of degrees \(p^2\) and \(p\).

We shall say that an indecomposable \(R_p\)-representation \(\Gamma\) of the group \(G\) has type \((\alpha,\beta,\gamma)\), if the decomposition of this representation over the field of rational numbers has the form

\[ \Gamma=\alpha\Gamma^{(1)}+\beta\Gamma^{(2)}+\gamma\Gamma^{(3)}. \]

For an arbitrary \(R_p\)-representation \(\Gamma\) of the group \(G\), we shall denote by \(n(\Gamma)\) the number of irreducible rational components of the representation \(\Gamma\), and by \(m(\Gamma)\) the maximum number of mutually equivalent irreducible rational components of this representation.

It is shown in \((^1)\) that the types of all indecomposable \(R_p(J_p)\)-representations of a cyclic group of order \(p^2\) \((p\ne 2)\) are exhausted by the vectors \((1,0,0)\); \((0,1,0)\); \((0,0,1)\); \((1,0,1)\); \((1,1,0)\); \((0,1,1)\); \((1,1,1)\); \((1,1,2)\).

In the present note the types of all indecomposable \(R_p\)-representations of any group of order \(mp^2\) are determined. In the second part of the paper a theorem is given proving the existence of infinite series of indecomposable \(Z\) \((R_p,J_p)\)-representations of a finite group all of whose irreducible rational components are mutually equivalent.

Theorem 1. Let \(G\) be a cyclic group of order \(mp^2\) \(((m,p)=1)\), and let \(\delta\) be the exponent to which the number \(p\) belongs modulo \(m\).

If \(p \ne 2\), then the types of indecomposable \(R_p\)-representations of the group \(G\) for \(\delta=\varphi(m)\) (\(\varphi\) is Euler’s function) are exhausted by the vectors:

\[ \begin{gathered} e_1=(1,0,0);\quad e_2=(0,1,0);\quad e_3=(0,0,1);\quad e_4=(1,1,0);\\ e_5=(0,1,1);\quad e_6=(1,0,1);\quad e_7=(1,1,1);\quad e_8=(1,1,2), \end{gathered} \tag{1} \]

for \(\delta=\varphi(m)/2\), by the vectors (1) and by the vectors:

\[ \begin{gathered} (2,1,1);\quad (1,2,1);\quad (2,1,2);\quad (1,2,2);\quad (2,2,2);\quad (2,3,3);\\ (3,2,3);\quad (3,2,2);\quad (2,3,2), \end{gathered} \tag{2} \]

and for \(\varphi(m)/\delta \ge 3\), by the vectors (1), (2), and

\[ (3,1,2);\quad (1,3,2);\quad (4,2,2);\quad (2,4,2);\quad (2,4,4);\quad (4,2,4). \]

For \(p=2\) the following types arise:

\[ \mathfrak{M}_1:\quad (1,0,0);\quad (0,1,0);\quad (0,0,1);\quad (1,1,0);\quad (1,0,1);\quad (0,1,1); \]
\[ (1,1,1)\qquad (\delta=\varphi(m)), \]

\[ \mathfrak{M}_1\cup\{(2,1,1),(1,2,1),(1,1,2),(1,2,2),(2,1,2),(2,2,2)\} \qquad (\delta\ge \varphi(m)/2). \]

Thus, every \(R_p\)-representation \(\Gamma\) of the group \(G\) is decomposable if at least one of the following conditions is satisfied:

\[ n(\Gamma)>10;\qquad m(\Gamma)>4\quad (p\ne 2), \]

\[ n(\Gamma)>6;\qquad m(\Gamma)>2\quad (p=2). \]

Let us note that from Theorem 1 it follows, in particular, that the cyclic group of order \(mp^2\) \(((m,p)=1)\) for \(p\ne 2\), \(\varphi(m)/\delta \ge 3\), has indecomposable \(R_p\)-representations with \(1,2,3,4,5,6,7,8\), and \(10\) rationally irreducible components.

The proof of Theorem 1 is based on the following propositions.

Lemma 1. Let \(S=\{\alpha_1e_1+\cdots+\alpha_8e_8\}\) be the set of all possible linear combinations of the vectors (1) with nonnegative integer coefficients \(\alpha_i\) \((i=1,\ldots,8)\), not all zero simultaneously, and let \(x=(\alpha,\beta,\gamma)\in S\) \((\alpha,\beta,\gamma>0)\).

If \(\alpha,\beta>\gamma\), then \(x-(1,1,0)\in S\). If \(\alpha,\beta<\gamma\), then \(x-(1,1,2)\in S\). If \(\alpha=\beta=\gamma\ge 2\), then \(x-(2,2,2)\in S\).

If \(\alpha>\beta=\gamma\), then \(x-(4,2,2)\in S\) for \(\alpha-\beta\ge 3,\ \beta\ge 2\); if \(\alpha>\beta=\gamma\) and \(\alpha-\beta=2\), then \(x-(2,2,2)\in S\) for \(\beta\ge 4,\ x-(2,1,1)\in S\) for \(\beta=3\); if \(\alpha>\beta=\gamma\) and \(\alpha-\beta=1\), then \(x-(2,1,1)\in S\) for \(\beta=2m+1\) and \(m\ge 1,\ x-(2,2,2)\in S\) for \(\beta=2m\) and \(m\ge 2\).

If \(\alpha<\beta=\gamma\) and \(\beta-\alpha\ge 2\), then \(x-(1,2,2)\in S\) for \(\alpha=2m+1\) and \(x-(2,4,4)\in S\) for \(\alpha=2m\) and \(m\ge 2\); if \(\alpha<\beta=\gamma\) and \(\beta-\alpha=1\), then \(x-(2,2,2)\in S\) for \(\beta=2m>2\) and \(x-(2,3,3)\in S\) for \(\beta=2m+1>3\).

If \(\alpha<\gamma<\beta\), then \(x-(1,3,2)\in S\) for \(\beta-\alpha>2\); if \(\alpha<\gamma<\beta\) and \(\beta-\alpha=2\), then \(x-(2,2,2)\in S\) for \(\alpha>2\) and \(x-(1,2,1)\in S\) for \(\alpha=2\).

Lemma 2. A \(J_p\)-representation of a finite group \(G\), equivalent over the field of \(p\)-adic numbers to a representation over the field of rational numbers, is \(J_p\)-equivalent to a representation over \(R_p\).

Lemma 2, as applied to a cyclic group \(G\), was used in \((^4)\) to prove the nonuniqueness of the decomposition of \(R_p\)-representations of the group \(G\) into a sum of indecomposable \(R_p\)-components. The general case of the lemma was proved by Heller \((^4)\).

Theorem 2. Let the group \(G\) be represented in the form of a semidirect product \(G=H\cdot F\), where \(H=(a)\) is a cyclic group of order \(p^\alpha\) \((1\le \alpha\le 2)\), and \(F\) is a group of order \(m\) \(((m,p)=1)\). Let \(M_1,\ldots,M_t\) \((t=4p+1\), if \(\alpha=2\), and \(t=3\), if \(\alpha=1)\) be a complete system of indecompo-

decomposable pairwise nonisomorphic \(H\)-\(J_p\)-modules, and \(M_i^*=g_1M_i+\cdots+g_mM_i\) (\(g_1,\ldots,g_m\) are elements of the group \(F\)) is the \(G\)-\(J_p\)-module induced by the \(H\)-\(J_p\)-module \(M_i\). Let \(e_1,\ldots,e_r\) be the minimal idempotents of the group ring \(J_pF\), to which correspond irreducible pairwise nonequivalent representations of the group \(F\) over the ring \(J_p\) (the number \(r\) is equal to the number of irreducible representations of the group \(F\) over the prime field of characteristic \(p\)). Then a complete system of indecomposable pairwise nonisomorphic \(G\)-\(J_p\)-modules is exhausted by the modules \((J_pG)e_jM_i\) \((j=1,\ldots,r;\ i=1,\ldots,t)\).

II. For a cyclic group \(G\) of order \(p^a\) (\(p\ne2\)), with \(a>2\), in (1) a series of indecomposable \(Z\)-representations of arbitrarily large degree has been given, in which only 3 rationally nonequivalent irreducible representations of the group \(G\) occur. Other series for a cyclic group with 4 irreducible rational nonequivalent representations were constructed in (5).

On the other hand, for a finite abelian group there exists only a finite number of indecomposable \(Z\)-representations with two distinct irreducible rational components (6).

Theorem 3. Let
\[ G:\quad a^p=1;\quad b^p=1;\quad c^p=1;\quad ab=ba;\quad ac=ca;\quad c^{-1}bc=ab \]
(\(p\ne2\); \(p\) prime). If \(p>3\), then there exist indecomposable \(Z(J_p,R_p)\)-representations of the group \(G\) of arbitrarily large degree, all irreducible rational components of which are equivalent to one and the same irreducible rational representation of this group of degree \(\varphi(p^2)\).

Uzhgorod
State University

Received
26 II 1964

REFERENCES

1. S. D. Berman, P. M. Gudivok, DAN, 145, No. 6, 1199 (1962).
2. A. Heller, J. Reiner, Ann. Math., 76, No. 1, (1962).
3. S. D. Berman, DAN, 152, No. 5, 1286 (1963).
4. A. Heller, Proc. Nat. Acad. Sci., 47, 1194 (1961).
5. A. Heller, J. Reiner, Ann. Math., 77, No. 2, 318 (1963).
6. V. S. Drobotenko, Abstracts of Reports and Communications of Uzhgorod State Univ., 37 (1963).