UDC 519-45

MATHEMATICS

S. D. BERMAN

GROUP ALGEBRAS OF COUNTABLE ABELIAN \(p\)-GROUPS

(Presented by Academician A. I. Mal'tsev, 29 IX 1966)

In the present note the problem of the isomorphism of group algebras of countable abelian \(p\)-groups is solved. The theorems formulated here were presented in the first part of the author’s report at the International Congress of Mathematicians (Moscow). Theorems 5 and 6 and one of the assertions of Theorem 2 have been partially published in \((^{1,2})\).

We shall agree to denote by \(GK\) the group algebra of a group \(G\) over a field \(K\). Deskins showed \((^3)\) that from the isomorphism of the group algebras \(GK\) and \(G_1K\) of two finite abelian \(p\)-groups \(G\) and \(G_1\) over a field of characteristic \(p\) there follows the isomorphism of the groups \(G\) and \(G_1\). Necessary and sufficient conditions for the isomorphism of the group algebras \(GK\) and \(G_1K\) of finite \(p\)-groups \(G\) and \(G_1\) over a field \(K\) whose characteristic is different from \(p\) were found in \((^{4,5})\).

Theorem 1. The group algebras \(GK\) and \(G_1K\) of two countable abelian \(p\)-groups \(G\) and \(G_1\) over a field \(K\) of characteristic \(p\) are isomorphic if and only if the groups \(G\) and \(G_1\) are isomorphic.

Let now \(p\) be a prime number, and let \(K\) be a field whose characteristic is different from \(p\). Denote by \(\xi_i\) a primitive root of unity of degree \(p^i\) \((i = 1, 2, \ldots)\).

We shall call a field \(K\) a field of the first kind relative to the prime \(p\) if \(K(\xi_j) \ne K(\xi_2)\) for some \(j > 2\). Otherwise we shall call \(K\) a field of the second kind (relative to the prime \(p\)).

Theorem 2. Let \(K\) be a field of the first kind relative to the prime \(p\). Every countable primary abelian \(p\)-group \(G\) belongs to exactly one of the following 9 types:

1) \(G\) is a direct product of cyclic groups with element orders unbounded in the aggregate;

2) \(G\) is a direct product of cyclic groups with element orders bounded in the aggregate;

3) \(G\) is the group \(p^\infty\);

4) \(G\) is a complete group whose direct decomposition contains at least two groups \(p^\infty\);

5) \(G\) is the direct product of the group \(p^\infty\) by a finite \(p\)-group \(H\) \((H \ne 1)\);

6) \(G\) is the direct product of a complete group of type 4) by a finite \(p\)-group \(H\) \((H \ne 1)\);

7) \(G\) is the direct product of a complete group by an infinite \(p\)-group without elements of infinite height and with element orders bounded in the aggregate;

8) \(G\) is a reduced \(p\)-group, and the subgroup \(P\) of elements of infinite height in \(G\) is finite and different from 1;

9) the subgroup \(P\) of elements of infinite height in \(G\) is infinite, and the orders of elements of the factor group \(G/P\) are not bounded in the aggregate.

If abelian \(p\)-groups \(G\) and \(G_1\) belong to different types, then the group algebras \(GK\) and \(G_1K\) are not isomorphic.

If \(G\) and \(G_1\) are groups of one and the same type \(n=1,3,4,8,9\), then the group algebras \(GK\) and \(G_1K\) are isomorphic.

Let \(G\) and \(G_1\) be groups of type 2), \(p^\alpha(p^{\alpha_1})\) the exponent of the group \(G(G_1)\), and \(p^\beta(p^{\beta_1})\) the greatest of the orders of those cyclic direct factors of the group \(G(G_1)\) that occur in a direct decomposition of the group \(G(G_1)\) a countable number of times. Let \(\eta(\eta_1)\) be a primitive \(p^\alpha(p^{\alpha_1})\)-th root of unity, and \(\varepsilon(\varepsilon_1)\) a primitive \(p^\beta(p^{\beta_1})\)-th root of unity. The group algebras \(GK\) and \(G_1K\) are isomorphic if and only if
\((K(\eta):K)=(K(\eta_1):K)\), \((K(\varepsilon):K)=(K(\varepsilon_1):K)\).

Let \(G\) and \(G_1\) be simultaneously groups of type 5) or 6): \(G=P\times H\), \(G_1=P_1\times H_1\) (\(P\) and \(P_1\) are groups \(p^\infty\) or complete groups of type 4), \(H\) and \(H_1\) are finite \(p\)-groups). The algebras \(GK\) and \(G_1K\) are isomorphic if and only if the algebras \(HK\) and \(H_1K\) are isomorphic.

Finally, let \(G\) and \(G_1\) be groups of type 7): \(G=P\times H\), \(G_1=P_1\times H_1\) (\(P\) and \(P_1\) are complete groups, \(H\) and \(H_1\) are countable \(p\)-groups of type 2)). The algebras \(GK\) and \(G_1K\) are isomorphic if and only if the algebras \(HK\) and \(H_1K\) are isomorphic.

Theorem 3. If \(p\ne2\), and \(K\) is a field of the second kind (relative to the prime \(p\)), then the group algebras \(GK\) and \(G_1K\) of any two countable abelian \(p\)-groups \(G\) and \(G_1\) are isomorphic.

Theorem 4. Let \(p=2\), and let \(K\) be a field of the second kind relative to \(p\). If \(K=K(\xi_2)\) (\(\xi_2\) is a primitive fourth root of unity), then \(GK\simeq G_1K\) for any two countable abelian \(2\)-groups \(G\) and \(G_1\).

If \(K\ne K(\xi_2)\), then the group algebra \(GK\) of an arbitrary countable abelian \(2\)-group \(G\) is isomorphic to the group algebra of a \(2\)-group of one of the following types:
\(G_1=(2,\ldots,2,\ldots)\) (the direct product of a countable number of cyclic groups of order \(2\));
\(G_2=(4,2,\ldots,2,\ldots)\);
\(G_3=(4,\ldots,4,\ldots)\);
\(G_4\) is the group \(2^\infty\);
\(G_5^s=G_4\times H_s\), where \(G_4\) is the group \(2^\infty\), \(H_s=(2,\ldots,2)\) is the direct product of \(s\) cyclic groups of order \(2\) (\(s\ge1\)).

The group algebras of \(2\)-groups of different types are not isomorphic.

1) \(GK\simeq G_1K\) if and only if \(G\simeq G_1\);

2) \(GK\simeq G_2K\) if and only if \(G\) contains no elements of infinite height and, in the decomposition of the group \(G\) into a direct product of cyclic groups, only a finite number of factors occur whose orders are greater than \(2\).

3) Let \(P\) be the subgroup of elements of infinite height of the group \(G\). \(GK\simeq G_3K\) if and only if, in the decomposition of the group \(G/P\) into a direct product of cyclic \(2\)-groups, infinitely many cyclic groups occur whose orders are greater than \(2\).

4) \(GK\simeq G_4K\) if and only if \(G\) is a complete \(2\)-group.

5) \(GK\simeq G_5^s\) if and only if \(G=P\times H\), where \(P\) is a complete \(2\)-group, and \(H\) is a finite \(2\)-group decomposing into a direct product of \(s\) cyclic groups.

Theorem 5. Let \(G\) and \(G_1\) be countable periodic abelian groups, and let \(K\) be an algebraically closed field whose characteristic does not divide the orders of the elements of the groups \(G\) and \(G_1\). Then the group algebras \(GK\) and \(G_1K\) are isomorphic.

Theorem 6. The group algebra \(GD\) of an arbitrary countable periodic abelian group \(G\) over the field of real numbers \(D\) is isomorphic to the real group algebra of one of the \(2\)-groups listed in the formulation of Theorem 4. Represent the group \(G\) in the form of a direct product
\(G=N\times P\times R\), where \(N\) is the subgroup of elements of odd order, \(P\) is a complete \(2\)-group, and \(R\) is a reduced \(2\)-group. If the subgroup \(N\times P\) is infinite and the group \(R\) is finite, then \(GD\simeq G_5^sD\), where \(s\) is the number of cyclic direct factors in the decomposition of the group \(R\).

\(GD\simeq G_3D\) if and only if the subgroup \(R\) is infinite and at least one of the following conditions is satisfied:

1) \(N \times P\) is an infinite group;
2) the subgroup \(R\) contains elements of infinite height;
3) \(N \times P\) is a finite group, and the group \(R\) decomposes into a direct product of cyclic groups, among which there are infinitely many groups whose orders are greater than 2.

\(GD \cong G_2D\) if and only if the group \(R\) decomposes into a direct product of a countable number of cyclic groups, \(R^2\) and \(N \times P\) are finite groups, and \(R \ncong G\) if \(N \times P = 1\).

\(GD \cong G_1D\) if and only if \(G \cong G_1\).

Theorem 7. Let \(G\) be a periodic group (of arbitrary cardinality), and let \(K\) be a field of characteristic zero. Every indecomposable \(GK\)-module is irreducible. The irreducible representations of the group \(G\) over the field \(K\) are in one-to-one correspondence with such sets \(E\) of idempotents of the algebra \(GK\) that:

1) the elements of \(E\) are the minimal idempotents of the group subalgebras \(HK\) of all possible finite subgroups \(H\) of the group \(G\); for each finite subgroup \(H \subseteq G\), the set \(E\) contains exactly one minimal idempotent \(e \in HK\);

2) any two idempotents from the set \(E\) are not orthogonal; irreducible representations \(\Gamma\) and \(\Gamma_1\) of the group \(G\) over the field \(K\) are equivalent if and only if the corresponding sets of idempotents \(E\) and \(E_1\) coincide.

Theorem 8. Let \(G\) and \(G_1\) be countable abelian \(p\)-groups; \(K\) a countable or finite field of characteristic \(p\); \(S\) and \(S_1\) the Sylow \(p\)-subgroups of the group multiplicative groups, respectively, of the algebras \(GK\) and \(G_1K\); \(P(P_1)\) the maximal complete subgroup of the group \(G(G_1)\); \(w(w_1)\) the ordinal type of the Ulm series of the group \(G/P\) \((G_1/P_1)\). If \(w=\gamma+1\) \((w_1=\gamma_1+1)\) is a transfinite number of the first kind, then denote by \(G^\gamma(G_1^{\gamma_1})\) the last factor of the Ulm series of the group \(G/P\) \((G_1/P_1)\). In this case the group \(G^\gamma(G_1^{\gamma_1})\) decomposes into a direct product of cyclic groups. Let \(p^\beta(p^{\beta_1})\) be the greatest of the orders of those cyclic direct factors of the group \(G^\gamma(G_1^{\gamma_1})\) which occur in the decomposition of the group \(G^\gamma(G_1^{\gamma_1})\) a countable number of times; \(H(H_1)\) the direct product of all cyclic direct factors of the group \(G^\gamma(G_1^{\gamma_1})\) whose orders do not exceed \(p^\beta(p^{\beta_1})\).

The groups \(S\) and \(S_1\) are isomorphic if and only if the following conditions are simultaneously satisfied:

1) if \(P \ne 1\), then \(P_1 \ne 1\);
2) \(w=w_1\);
3) if \(w=w_1=\gamma+1\) and \(K\) is a countable field, then the groups \(G^\gamma\) and \(G_1^\gamma\) have one and the same exponent \(p^\alpha\), or the orders of the elements of these groups are unbounded. If \(w=\gamma+1=w_1\) and \(K\) is a finite field, then the orders of the elements of the groups \(G^\gamma\) and \(G_1^\gamma\) are simultaneously unbounded or bounded; moreover, in the latter case the exponents of the groups \(G^\gamma\) and \(G_1^\gamma\) coincide, and the finite groups \(G^\gamma/H\) and \(G_1^\gamma/H_1\) are isomorphic.

Uzhgorod State
University

Received
19 IX 1966

CITED LITERATURE

\(^{1}\) S. D. Berman, Dokl. i soobshch. Uzhgorodsk. univ., ser. phys.-math. sciences, No. 3, 56 (1960).
\(^{2}\) S. D. Berman, ibid., No. 4, 76 (1961).
\(^{3}\) W. E. Deskins, Duke Math. J., 23, 35 (1956).
\(^{4}\) S. Perlis, G. Walker, Trans. Am. Math. Soc., 68, 420 (1950).
\(^{5}\) S. D. Berman, DAN, 91, No. 2, 185 (1953).