Reports of the Academy of Sciences of the USSR

1. Vol. 145, No. 6

MATHEMATICS

S. D. BERMAN and P. M. GUDIVOK

ON INTEGRAL REPRESENTATIONS OF FINITE GROUPS

(Presented by Academician P. S. Novikov on 29 III 1962)

Higman ((^{1})) showed that a finite group (G) and its Sylow (p)-subgroup simultaneously have a finite or an infinite number of indecomposable representations over an arbitrary field (K) of characteristic (p). The same result holds for representations of finite groups over the ring of integral (p)-adic numbers (I_p). Since an abelian (p)-group of type ((p,p)), according to ((^{2})), has over the ring (I_p) indecomposable representations of arbitrarily high degree, the question of finite groups with a finite number of indecomposable representations over this ring reduces to the analogous question for cyclic primary (with respect to (p)) groups. Diederichsen ((^{3})) showed that a group of order (p) has over the ring (I_p) and over the ring of rational integers (Z) a finite number of indecomposable representations, and the degrees of these representations do not exceed the order of the group. A. V. Roiter ((^{4})) proved the same result for representations of the cyclic group of order (4) over the rings (I_p) and (Z). (The series of indecomposable representations of arbitrarily large degree given in ((^{3})) turned out to be erroneous.)

In the present note the question of finite groups admitting only a finite number of indecomposable representations over the ring (I_p) is completely solved, and a number of theorems on representations of finite groups over the ring of rational integers (Z) are given.

Let (G=(a)) be a cyclic group of order (p^2) ((p) prime). Denote by (\xi) and (\varepsilon), respectively, primitive roots of degree (p^2) and (p) of (1). Form the rings (I_p[\xi]) and (I_p[\varepsilon]) and choose in them (I_p)-bases:

[
1,\ \xi,\ \ldots,\ \xi^{\varphi(p^2)-1};
\tag{1}
]

[
1,\ \varepsilon,\ \ldots,\ \varepsilon^{\varphi(p)-1}.
\tag{2}
]

Let (\widetilde{\xi}) (respectively (\widetilde{\varepsilon})) be the matrix corresponding to the operator of multiplication by (\xi) (respectively by (\varepsilon)) in the basis (1) (respectively (2)). The group (G) has 3 irreducible representations over the ring (I_p): (a \to 1); (a \to \widetilde{\varepsilon}); (a \to \widetilde{\xi}).

If a representation

[
a \to
\begin{pmatrix}
\widetilde{\xi} & A\
0 & \widetilde{\varepsilon}
\end{pmatrix}
]

is given, then to each column

[
\begin{pmatrix}
a_{1j}\
\vdots\
a_{kj}
\end{pmatrix}
]

((k=\varphi(p^2)); (\varphi) is Euler’s function) one may put in correspondence the element
(a_{1j}+\cdots+a_{kj}\xi^{k-1}) of the ring (I_p[\xi]).

Theorem 1. All indecomposable representations of the group (G=(a)) ((a^{p^2}=1;\ p>2)) over the ring (I_p), up to equivalence, are exhausted by the following series:

[
\begin{gathered}
1)\ 1;\qquad
2)\ \widetilde{\varepsilon};\qquad
3)\ \widetilde{\xi};\qquad
4)\
\begin{pmatrix}
\widetilde{\xi} & (t)\
0 & \widetilde{\varepsilon}
\end{pmatrix};\qquad
5)\
\begin{pmatrix}
\widetilde{\xi} & (1)\
0 & 1
\end{pmatrix};\qquad
6)\
\begin{pmatrix}
\widetilde{\varepsilon} & (1)\
0 & 1
\end{pmatrix};
\[6pt]
7)\
\begin{pmatrix}
\widetilde{\xi} & (t) & 0\
0 & \widetilde{\varepsilon} & (1)\
0 & 0 & 1
\end{pmatrix};\qquad
8)\
\begin{pmatrix}
\widetilde{\xi} & 0 & (1)\
0 & \widetilde{\varepsilon} & (1)\
0 & 0 & 0
\end{pmatrix};\qquad
9)\
\begin{pmatrix}
\widetilde{\xi} & (t) & (1)\
0 & \widetilde{\varepsilon} & 0\
0 & 0 & 1
\end{pmatrix},
\quad t=1,\ldots,\varphi(p)-1;
\[6pt]
10)\ \Gamma_t(1)=
\begin{pmatrix}
\widetilde{\xi} & (t) & 0 & (1)\
0 & \widetilde{\varepsilon} & (1) & 0\
0 & 0 & 1 & 0\
0 & 0 & 0 & 1
\end{pmatrix}
\quad (t=2,\ldots,\varphi(p)-1).
\end{gathered}
]

Here ((t)) denotes a rectangular matrix with zero columns except for the last one, whose last column corresponds to the element ((1-\xi)^{t-1}\in I_p[\xi])*.

For (p=2) the first 9 series of Theorem 1 give the indecomposable (Z)-representations of the cyclic group of order 4, indicated by A. V. Roiter ((^4)). The degrees of the representations of series 1)—9) do not exceed the order of the group, while series 10) gives representations of degree (p^2+1) (it appears only for (p>2)).

Theorem 2. The cyclic group (H=(a)) of order (p^3) ((p) an arbitrary prime) has, over the ring (I_p), indecomposable representations of arbitrarily large degree that can be realized by integer matrices.

Let (\eta) be a primitive root of degree (p^3) of 1; let (\eta) be the matrix corresponding to the operator of multiplication by (\eta) in the (I_p)-basis (1,\eta,\ldots,\eta^{\varphi(p^3)-1}) of the ring (I_p[\eta]), and let (m) be an arbitrary natural number.

Then, for (p>2), the representations of the following series will be indecomposable:

[
a\to
\begin{pmatrix}
(\widetilde{\eta}\times E_m) & ((t)\times E_m) & ((1)\times D_m)\
0 & (\widetilde{\xi}\times E_m) & ((t)\times E_m)\
0 & 0 & (\widetilde{\varepsilon}\times E_m)
\end{pmatrix},
\tag{3}
]

where (\times) is the sign of Kronecker multiplication; (D_m) is a Jordan box of size (m) with ones on the main diagonal; ((t)) denotes a matrix in which only the last column is nonzero, and this column, depending on the position of the matrix ((t)) in the matrix (3), corresponds either to the element ((1-\eta)^{t-1}\in I_p[\eta]), or to the element ((1-\xi)^{t-1}\in I_p[\xi]); (E_m) is the identity matrix.

In the representations of series (3) only three irreducible components (\widetilde{\eta}), (\widetilde{\xi}), and (\widetilde{\varepsilon}) of the group (H) participate. If (p=2), then in order to construct an infinite series of indecomposable representations of the group (H), it is necessary to involve all irreducible components of this group over the ring (I_2).

Theorem 3. A finite group (G) has a finite number of indecomposable representations over the ring of integral (p)-adic numbers (I_p) if and only if a Sylow (p)-subgroup of this group is a cyclic group whose order is not divisible by (p^3).

Theorem 4. A finite group (G) whose order is divisible by the cube of a prime number possesses indecomposable representations of arbitrarily large degree over the ring of rational integers (Z).

Theorem 5. The number of indecomposable representations of a finite cyclic group over the ring (Z) is finite if and only if its order is free of cubes.

Theorem 6. A finite group with square-free order (not necessarily commutative) has only a finite number of indecomposable representations over the ring (Z) ((Z) is the ring of rational integers).

In Reiner’s paper ((^5)) the question was posed: is the decomposition of a representation of a finite group over the ring (R_p) of (p)-adic rational numbers into a sum of indecomposable representations over this ring always unique?

A negative answer to this question already arises for (p)-adic representations of cyclic groups of square-free order. The indecomposable (I_p)-components of a representation (\Gamma) over (R_p) determine this representation up to (R_p)-equivalence, but these components may be combined in different ways into (R_p)-components of the representation (\Gamma). The integral forms of the representations of series 10) for the cyclic group of order (p^2) ((p>2)) make it possible to give a negative solution to Reiner’s second problem ((^6)): can nonequivalent (Z)-representations of a finite group (G) become equivalent after extending the ring (Z) to the ring of all integral elements of some field of algebraic numbers? Let (p=4m+1),

* In particular, ((1)) denotes a matrix of the form (\binom{1}{0}).

and (\alpha) is a quadratic nonresidue modulo (p). We form the (Z)-representation (\Gamma_t(\alpha)) of the cyclic group ((a)) ((a^{p^2}=1)) by replacing the column

[
\begin{pmatrix}
1\
0\
\vdots\
0
\end{pmatrix}
]

by the column

[
\begin{pmatrix}
\alpha\
0\
\vdots\
0
\end{pmatrix}
]

(see series 10). Then the representations (\Gamma_t(1)) and (\Gamma_t(\alpha)) are not equivalent over the ring (Z), but are already equivalent over the ring (Z(\varepsilon)), where (\varepsilon^p=1).

Moreover, there is a (Z)-equivalence
[
\Gamma(1)+\Gamma(\alpha)=\Gamma(1)+\Gamma(1),
]
and hence, for representations of finite groups over the ring of rational integers, the cancellation condition is not satisfied.

The authors express their deep gratitude to D. K. Faddeev and I. R. Shafarevich for their constant interest in their work and for a number of valuable comments.

Uzhgorod State
University

Received
21 III 1962

REFERENCES

1. D. G. Higman, Duke Math. J., 21, No. 2, 377 (1954).
2. Z. I. Borevich, D. K. Faddeev, Vestn. LGU, ser. matem., issue 2, No. 7, 72 (1959).
3. F. Diederichsen, Abh. Math. Sem. Hamburg, 14, 357 (1938).
4. A. V. Roiter, Vestn. LGU, ser. matem., 19, 4, 65 (1960).
5. J. Reiner, Bull. Am. Math. Soc., 67, 4, 365 (1961).
6. J. Reiner, Illinois. J. Math., 4, No. 4, 640 (1960).