S. D. BERMAN

ON INTEGRAL REPRESENTATIONS OF FINITE GROUPS

(Presented by Academician P. S. Novikov on 30 V 1963)

We shall use the following notation: \(Z\) is the ring of rational integers; \(R_p\) is the ring of \(p\)-integral rational integers; \(J_p\) is the ring of integral \(p\)-adic integers; \(Q\) is the field of rational numbers; \(\Omega_p\) is the field of \(p\)-adic numbers; \(R(G,T)\) is the group ring of the group \(G\) over the ring \(T\).

In \((^1)\) the following theorem was established:

Theorem 1. A finite group \(G\) has a finite number of indecomposable \(Z\)-representations if and only if its order is cube-free and all Sylow \(p\)-subgroups of this group are cyclic.

If the group \(G\) satisfies the conditions of Theorem 1, then, generally speaking, there exist indecomposable \(Z\)-representations of this group that are not realized in ideals of the integral group ring \(R(G,Z)\). For example, as shown in \((^2)\), a cyclic group of order \(p^2\) \((p \ne 2)\) has an indecomposable \(Z\)-representation of degree \(p^2 + 1\).

We shall say that a finite group \(G\) has property \((J)\) if all indecomposable \(Z\)-representations of this group are realized in ideals of the group ring \(R(G,Z)\).

In the present note the class of finite groups satisfying condition \((J)\) is completely determined. In the second part of the paper a theorem on induced \(Z\)-representations is given, generalizing the known theorem on the indecomposability of the integral group ring \(R(G,Z)\) of an arbitrary finite group \(G\), and a theorem on the number of irreducible \(J_p\)-representations of an arbitrary \(p\)-group.

I. Theorem 2. A finite cyclic group \(G\) has property \((J)\) if and only if its order is either square-free or equal to \(4\).

Theorem 3. The following conditions are equivalent:

1) The group \(G\) has property \((J)\).

2) The order of the group \(G\) is either square-free, or is equal to \(4p_1 \cdots p_s\) \((p_1,\ldots,p_s\) are odd pairwise distinct primes), and in the latter case the group \(G\) simultaneously satisfies the following conditions: a) \(G\) is an extension of a normal divisor \(H\) of order \(p_1 \cdots p_s\) by a cyclic group of order \(4\); b) \(G\) contains no cyclic subgroups of order \(4p_i\) \((i=1,\ldots,s)\).

From Theorem 3, in particular, it follows that for a group \(G\) of odd order property \((J)\) holds if and only if the order of the group is square-free.

Theorem 3 also admits the following formulation:

A group \(G\) has property \((J)\) if and only if all Sylow subgroups of the group \(G\) are cyclic and the order of each cyclic subgroup of the group \(G\) is either square-free or equal to \(4\).

The proof of the necessity of the conditions of Theorem 3 is based on Theorem 2. If the group \(G\) contains a cyclic subgroup \(H\) whose order is divisible by \(p^2\) (\(p\) an odd prime) or by \(4p\), then the subgroup \(H\) has an indecomposable \(Z\)-representation \(\Gamma\) that is not realized in ideals of the ring \(R(H,Z)\), and with the aid of \(\Gamma\) one can construct an indecomposable \(Z\)-representation of the group \(G\) which likewise cannot be obtained with the aid of ideals of the ring \(R(G,Z)\).

The proof of sufficiency is technically more complicated and is based on the following facts, which are of independent interest.

Theorem 4. Let \(G\) be a group with cyclic Sylow \(p\)-subgroups (for all \(p\mid (G:1)\)). If \(\Gamma\) is a \(Z\)-representation of the group \(G\), all irreducible \(Q\)-components of which are mutually equivalent, then \(\Gamma\) (over \(Z\)) decomposes into a sum of irreducible \(Z\)-components:
\[ \Gamma=\Gamma_1+\cdots+\Gamma_m . \]

Lemma 1. Let \(G\) be a group of square-free order:
\[ G:\quad a_1^{p_1}=1,\ldots,a_s^{p_s}=1;\quad a_i a_j=a_j a_i\quad (i,j=1,\ldots,s); \]
\[ b_1^{q_1}=1,\ldots,b_r^{q_r}=1;\quad b_i b_j=b_j b_i\quad (i,j=1,\ldots,r); \]
\[ b_i^{-1}a_j b_i=a_j^{\mu}\quad (p_1,\ldots,p_s;\ q_1,\ldots,q_r\text{ are distinct primes}). \]
Let
\[ \Gamma:\ a\to \tilde{\varepsilon};\ b\to B \quad (a=a_1\cdots a_s;\quad b=b_1\cdots b_r) \]
be an irreducible \(Q\)-representation of the group \(G\), where \(\tilde{\varepsilon}\) is the matrix of the operator of multiplication by a primitive root \(\varepsilon\) of degree \(p_1\cdots p_r\) in the field \(Q(\varepsilon)\). Suppose that the elements \(b_1,\ldots,b_r\) are numbered so that the elements \(b_1,\ldots,b_k\) exhaust all elements \(b_i\) \((1\le i\le r)\) that simultaneously satisfy the conditions
\[ [b_i,a_1]\ne 1;\qquad [b_i,a_j]=1\quad (j=2,\ldots,r). \]
Then there exist exactly \(q_1\cdots q_k\) pairwise inequivalent \(R_p\)-representations of the group \(G\) that are equivalent over \(Q\) to the representation \(\Gamma\).

Lemma 2. Let the order of the group \(G\) be cube-free and let all Sylow \(p\)-subgroups of this group be cyclic. Let the \(G-J_p\)-module \(M\) be represented as a direct sum of irreducible \(G-J_p\)-modules:
\[ M=M_1+\cdots+M_s . \]
If the modules
\[ \Omega_p M_1,\ldots,\Omega_p M_s \]
are \(G-\Omega_p\)-isomorphic, then every \(G-J_p\)-automorphism \(\theta\) of the module \(M\) can be represented in the form of a product
\[ \theta=\theta'\theta_1\cdots\theta_k, \]
where \(\theta'\) is a diagonal automorphism, and \(\theta_i\) \((i=1,\ldots,k)\) are elementary triangular automorphisms of this module.

We note that Theorem 4 and Lemma 2 play an essential role in the author’s proposed proof of Theorem 1.

II. It is known that the integral group ring \(R(G,Z)\) is indecomposable into a direct sum of left (right) ideals. This fact may be formulated as follows: the \(Z\)-representation of the group \(G\) induced by the unit character of the unit subgroup of the group \(G\) is indecomposable.

Theorem 5. Let \(G\) be an arbitrary finite group, \(H\) a subgroup of the group \(G\), \(\chi\) a linear character of the subgroup \(H\), and let \(\Gamma\) be the representation of the group \(G\) over the ring \(Z[\chi]^*\), induced by this linear character. Then the representation \(\Gamma\) is indecomposable over the ring \(\bar Z\) of all algebraic integers.

In particular, the transitive permutation representation of the group \(G\), regarded as a \(Z\)-representation of the group \(G\), is indecomposable.

The following theorem, by its methods of proof, is closely connected with Lemma 1.

Theorem 6. Let \(G\) be a \(p\)-group \((p\ne 2)\); let \(\bar{\Gamma}\) be an absolutely irreducible representation of the group \(G\); let \(\chi\) be the character of the representation \(\bar{\Gamma}\); and let \(\Gamma\) be an irreducible representation of the group \(G\) over the field \(\Omega_p\), corresponding to the representation \(\bar{\Gamma}\). Let
\[ T=\Omega_p(\varepsilon) \]
be the smallest splitting field of the circle in which the representation \(\bar{\Gamma}\) can be written monomially \(\bigl(T\supset \Omega_p(\chi)\bigr)\).

Then there exist at least
\[ m=(T:\Omega_p(\chi)) \]
pairwise inequivalent \(J_p\)-representations (and \(R_p\)-representations) of the group \(G\) that are equivalent over \(\Omega_p\) to the representation \(\Gamma\).

Let us point out that it is easy to indicate examples where \(m>1\). For example, for the group
\[ G:\quad a^{p^2}=1;\quad b^p=1;\quad b^{-1}ab=a^{1+p}\quad (p\ne 2) \]
there exists an absolutely irreducible representation \(\bar{\Gamma}\) of degree \(p\), for which \(m=p\).

Uzhgorod
State University

Received
16 V 1963

References

1. S. D. Berman, P. M. Gudivok, Reports and Communications of Uzhgorod University, series phys.-math., No. 5, 74 (1962).
2. S. D. Berman, P. M. Gudivok, Dokl. Akad. Nauk SSSR, 145, No. 6, 1199 (1962).

* The ring \(Z[\chi]\) is obtained by adjoining to the ring \(Z\) all values of the character \(\chi\).