Key words: finite groups, finite rings, varieties of groups and rings.

AMS Mathematical Subject Classification: 20E10, 20D99, 17A30, 08B20.

1 Introduction↩︎

Let \(A\) be a finite algebra of a fixed signature. The variety of algebras \({\mathfrak{V}}=var\,A\) generated by \(A\) consists of all algebras of the same signature satisfying every identity that holds in \(A\). A basic theorem of G. Birkhoff [1] says that \({\mathfrak{V}}\) is the smallest class of algebras containing \(A\) and closed under taking subalgebras, homomorphic images, and Cartesian products.

The order of every \(n\)-generated algebra in the variety \(\mathfrak{V}\) is bounded by the order of the \(n\)-generated \(\mathfrak{V}\)-free algebra \(F_n\), and \(|F_n|\) is bounded above by the double-exponential function \(|A|^{|A|^n}\), where \(|A|\) is the order of the algebra \(A\). It follows that \(\limsup_{n\to\infty}\sqrt[n]{\log |F_n|}\le |A|\). We denote the left-hand side of this inequality by \(pt(A)\) and call it the potential of the finite algebra \(A\).

For example, if \(G\) is a nontrivial finite group, then \(pt(G) =1\) if \(G\) is nilpotent and \(pt(G) \ge 2\) otherwise ([2], 24.52). The same alternative holds for finite-dimensional (nonassociative) algebras over finite fields [3].

As early as in 1937, it was noticed by B.H. Neumann [4] that the Birkhoff’s estimate can be improved in many cases. Our goal is to find necessary and sufficient conditions on \(A\) under which the potential \(pt(A)\) attains its maximal possible value, namely conditions equivalent to the equality \(pt(A) = |A|\), where \(A\) is either a finite group or a finite, not necessarily associative, ring.

Recall that a nontrivial group (ring) \(A\) is called monolithic if the intersection of all nontrivial normal subgroups (ideals) of \(A\) is nontrivial. This intersection is the monolith of the monolithic group (respectively, ring) \(A\).

Many nonsimple groups satisfy condition (*); for example, the symmetric groups \(S_n\) for \(n\ge 5\) and any wreath product of a non-cyclic finite simple group and a nontrivial finite group.

A (not necessarily associative) ring \(A\) is a prime ring if the product of any two nonzero ideals of \(A\) is nonzero. The following is the analogue of Theorem 1.

Classical results in ring theory imply that every finite prime associative ring is simple ([5], Section 98). By contrast, the following construction gives a nonsimple prime algebra over a field with basis \((e,f)\) and multiplication rules \(e^2 = f\), \(ef=f\), \(fe=0\), and \(f^2=f\). It has only one one-dimensional ideal \(\langle f\rangle\).

2 Sufficiency in Theorem 1↩︎

If \(k=1\), then the assertion is immediate. Arguing by induction, assume now that \(k\ge 2\) and \(\phi: S\to G\) is an epimorphism. If, for example, \(\phi(S\cap H_k)=1\), then \(\phi\) factors through the projection \(\pi_k: S\to H_1\times \dots\times H_{k-1}\) since \(S\cap H_k\) is the kernel of \(\pi_k\), and so \(G\) is an epimorphic image of a subgroup of \(H_1\times \dots\times H_{k-1}\), contrary to the inductive hypothesis. Otherwise the nontrivial normal subgroups \(\phi(S\cap H_1)\) and \(\phi(S\cap H_2)\) of \(G\) commute elementwise. This is impossible since the monolith \(M\) of \(G\) is non-abelian. This contradiction proves the lemma.

Let \(G\) be a finite monolithic group with a non-abelian monolith \(M\) and \(P=P_k\) be the direct product of isomorphic copies \(G_i\) of the group \(G\), \(i=1,\dots, k\), and and \(R = P\times H\), where the order of the group \(H\) is less than the order of \(G\). Denote by \(\pi_i\) the projections of \(R\) onto \(G_i\). Suppose \(S\) is subdirectly embedded in the group \(R\). (Recall that a subgroup \(S\) of a direct product is called a subdirect product of the factors if it projects onto each of them.) Suppose that \(n\) elements \(x_1,\dots,x_n\) are chosen in \(S\) (not necessarily distinct). Their projections to \(G_i\) are denoted by \(x_{i1},\dots,x_{in}\), respectively.

We induct on the number of factors \(k\) in the direct product \(P\). If \(k=0\), the assertion is trivial since then \(|S|\le |H|<|G|\) and no epimorphism \(S\to G\) exists. So we assume that \(k\ge 1\).

If \(S\cap G_k = \{1\}\), then we consider the projection \(\alpha\) of \(S\) to \(G_1\times\dots\times G_{k-1}\times H\) and observe that \(\alpha\) is an isomorphism between \(S\) and its \(\alpha\)-image. By the inductive hypothesis, the statement of the lemma holds for the epimorphism \(\phi\alpha^{-1}: S\to G\), and therefore \(\phi\alpha^{-1} = \psi\pi_i\) for some \(i\le k-1\) and an isomorphism \(\psi: G_i\to G\) mapping the elements \(x_{i1},\dots,x_{in}\) to \(y_1,\dots,y_n\), respectively. Then \(\phi = \psi\pi_i\alpha = \psi\pi_i\) as required. Hence we may assume from now on that \(S\cap G_i \ne 1\) for every \(i\le k\).

The intersections \(S\cap G_i\) are normal in \(G_i\) for every \(i\le k\), because \(\pi_i(S) = G_i\). Hence the subgroup \(S\) contains the product \(N\) of all the monoliths \(M_1,\dots, M_k\) of the factors \(G_1,\dots, G_k\), and each \(M_i\) is a minimal normal subgroup in \(S\).

The kernel \(K=\ker\phi\) cannot contain \(N\), since otherwise \(G\) would be a homomorphic image of a subgroup of a direct product \((G/M)^k\times H\), but this is impossible by Lemma 1. Without loss of generality, suppose \(M_k\) is not contained in \(K\).

Denote by \(Q\) the intersection \(S\cap (G_1\times\dots\times G_{k-1}\times H)\); it is the kernel of the projection of \(S\) onto \(G_k\), and therefore \(S/Q\simeq G_k\). If the normal subgroup \(Q\) of \(S\) is not contained in \(K\), then the nontrivial normal \(\phi\)-images of \(Q\) and \(M_k\) in \(G\) commute elementwise since \(Q\cap M_k=1\), a contradiction, because the monolith of \(G\) is not abelian. Hence we have \(Q\le K\).

Since the kernel \(Q\) of \(\pi_k\) is contained in the kernel \(K\) of \(\phi\), the isomorphism \(\psi= \phi \pi_k^{-1}: G_k\to G\) is well defined; that is, \(\phi = \psi\pi_k\). The equalities \(\pi_k(x_i) = x_{ki}\) and \(\phi(x_i) = y_i\) imply the equalities \(\psi(x_{ki}) = y_i\) for \(i=1,\dots, n\), and the lemma is proved.

For each maximal proper subgroup \(H\le G\), we have \(|H|\le |G|/2\) and the number of \(n\)-tuples of elements from \(H\) is equal to \(|H|^n\). If \(s\) is the number of proper subgroups of \(G\), then the number of \(n\)-tuples generating \(G\) is at least \(|G|^n-s(|G|/2)^n=|G|^n(1-O(2^{-n}))\).

Call two generating \(n\)-tuples of \(G\) equivalent if one can be mapped to the other by an automorphism of \(G\). Then there are at least \(N=\frac{|G|^n}{|{\rm Aut}\,G|}\left(1-O(2^{-n})\right)> c|G|^n\) pairwise inequivalent \(n\)-tuples in \(G\), for some \(c>0\). Choose a maximal set of pairwise inequivalent \(n\)-tuples and denote it by \(Y\). Enumerate the \(n\)-tuples in \(Y\) by the integers \(1,\dots,N\).

Let \(G_j\) be an isomorphic copy of \(G\) generated by \(x_{1j},\dots, x_{nj}\), where \((x_{1j},\dots, x_{nj})\) is the \(j\)th \(n\)-tuple in \(Y\), \(j=1,\dots,N\). Denote by \(M_j\) the copy of the monolith \(M\) in \(G_j\). In the direct product \(P\) of all \(G_j\), we choose the elements \[\begin{align} x_1&=& (x_{11},\dots,x_{1N}) \\ x_2 &=& (x_{21},\dots,x_{2N}) \\ \dots &=& \dots\dots\dots\dots\dots\\ x_n &=& (x_{n1},\dots,x_{nN}) \end{align}\] where the coordinates of the \(j\)th column form the \(j\)th generating \(n\)-tuple of \(G\).

Let \(B\) be the subgroup of \(P\) generated by the elements \(x_1,\dots, x_n\). We will prove that \(B\) contains the product \(M_1\times\dots\times M_N\). It follows that \(t(n) \ge N > c |G|^n\), as required.

Arguing by contradiction, assume, without loss of generality, that \(B\) does not contain \(M_N\). Since \(M_N\) is the monolith of \(G_N\), it follows that \(B\cap M_N = 1\), because this normal subgroup of the subdirect product \(B\) must be normal in the factor \(G_N\). Hence \(B\) admits a natural subdirect embeddeding in the product \(G_1\times\dots\times G_{N-1}\times H\), where \(H= G_N/M_N\).

By Lemma 2, the projection \(\pi_N: B\to G_N\) factors as \(\psi\pi_i\), where \(i\le N-1\) since \(|H|<|G|\), and \(\psi\) is an isomorphism \(G_i\to G_N\), with \(\psi(x_{ji}) = x_{jN}\) for \(j=1,\dots, n\). In other words, there is an automorphism of \(G\) mapping the \(n\)-tuple \((x_{1i},\dots, x_{ni})\) to the \(n\)-tuple \((x_{1N},\dots, x_{nN})\), which is impossible by the definition of the set \(Y\) and the definition of the equivalence of \(n\)-tuples. This contradiction proves the lemma.

It follows from Lemma 3 that for all sufficiently large \(n\) the variety \(var\; G\) contains an \(n\)-generated group \(B\) of order at least \(|M|^{c|G|^n}\). Therefore \(\log|F_n|> c_1 |G|^n\) for some \(c_1>0\) and every \(n\ge 1\), as required. Birkhoff’s inequality implies \(\log|F_n|< c_2 |G|^n\) for some \(c_2>0\). This proves the sufficiency part of Theorem 1.

3 Necessity in Theorem 1↩︎

We now assume that \(pt(G)=|G|\), where \(G\) is a finite group and \(\mathfrak{V}= var\; G\). Then \(G\) is nontrivial, since \(pt(\{1\})=0\).

Clearly, \(pt(G)\ge\max (p_1,p_2)\) since \(\mathfrak{V}\) contains both \(\mathfrak{V}_1=var\; G_1\) and \(\mathfrak{V}_2=var\; G_2\). On the other hand, the \(n\)-generated \(\mathfrak{V}\)-free group \(F_n\) is a subdirect product of the free groups \(F_n'\) and \(F_n''\) in the varieties \({\mathfrak{V}}_1\) and \({\mathfrak{V}}_2\) ([2], 15.82), and hence \(|F_n|\le |F'_n||F''_n|\). It remains only to note that, for any sequences of positive integers \(\{a'_n\}_{n=1}^{\infty}\) and \(\{a''_n\}_{n=1}^{\infty}\), we have

\[\limsup_{n\to\infty}\sqrt[n]{\log (a'_na''_n)}= \limsup_{n\to\infty}\sqrt[n]{\log a'_n+\log a''_n}\] \[= \max (\limsup_{n\to\infty}\sqrt[n]{\log a'_n}, \; \limsup_{n\to\infty}\sqrt[n]{\log a''_n}).\]

If the group \(G\) is not monolithic, there are two nontrivial normal subgroups \(N_1\) and \(N_2\) in \(G\) such that \(N_1\cap N_2 = 1\). Then \(G\) is a subdirect product of the factor groups \(G_1=G/N_1\) and \(G_2 = G/N_2\), and \(var\; G = var\;\{G_1,G_2\}\). Thus \(p_1\le |G_1|<|G|\) and \(p_2\le |G_2|<|G|\) in Lemma 3, whence \(pt(G)<|G|\), a contradiction. Therefore \(G\) must be monolithic.

It remains to prove that \(pt(G)<|G|\) if there is a nontrivial abelian normal subgroup \(N\) in a finite group \(G\). Since finite free groups in the variety \(\mathfrak{V}=var\; G\) are isomorphic to some subgroups of the direct powers of \(G\) ([2], 15.4), we must substantially reduce the number of direct factors (in comparison with groups having non-abelian monolith). To this end, we construct subdirect products with far more non-equivalent homomorphisms to \(G\) than standard projections onto the direct factors.

Let \(M\) be an abelian normal subgroup in a finite group \(H\). Then \(z^{-1}mz\) is well defined for every \(m\in M\) and \(z\in H/M\), and it belongs to \(M\). For each coset \(z\in H/M\), choose a representative \(u(z)\in z\) and denote \(U=\{u(z)\mid z\in H/M\}\). Then \(u(y)u(z) = f(y,z)u(yz)\) for some \(f(y,z)\in M\). Choose also \(n\ge 1\) and denote by \(X=(x_1,\dots,x_n)\) an \(n\)-tuple of elements from \(H\). We shall work with quadruples \(Q=(H,M,U,X)\).

We call two quadruples \(Q=(H,M,U,X)\) and \(Q'=(H',M', U',X')\) similar if there exist isomorphisms \(M\to M'\) and \(H/M\to H'/M'\) such that

• \((z')^{-1}m'z' = (z^{-1}mz)'\) for \(z\in H/M\) and \(m\in M\). (Here primes denote the images of the elements \(m\) and \(z\) under the group isomorphisms \(M\to M'\) and \(H/M\to H'/M'\).)

• If a component \(x_i\) of \(X\) belongs to a coset \(z\in H/M\), then \(x'_i\in z'\in H'/M'\).

The second requirement means that \(x_i= g_iu(z)\) and \(x'_i =h_iu'(z')\) for some \(g_i\in M\), \(h_i\in M'\).

If, in addition to similarity, we have \(f'(y',z') = f(y,z)'\) and \(h_i=(g_i)'\) for all \(y, z, i\), then we say that \(Q\) and \(Q'\) are isomorphic quadruples. Similarity and isomorphism of quadruples are both equivalence relations. We take quadruples up to isomorphism.

The sum of two similar quadruples \(Q=(H,M,U,X)\) and \(Q'=(H',M',U',X')\) is constructed as follows. First form the direct product \(P=H \times H'\), containing \(L= M \times M'\). The isomorphism of the factor groups \(H/M\) and \(H'/M'\) defines the diagonal \(D_0\) of the direct product \(H/M\times H'/M'\). Let \(D\) be the inverse image of \(D_0\) in \(P\) under the natural mapping \(P\to P/L\). Thus the group \(D/L\) isomorphically projects onto \(H/M\), and the set of representatives \(U_0\) of the cosets of \(L\) in \(D\) consists of the pairs \(v(z_0)=(u(z),u'(z'))\), where \(z\in H/M\). The factor group \(D/L\) acts by conjugation on \(L\) componentwise: \(z_0^{-1}(m,m')z_0 = (z^{-1}mz, (z')^{-1}m'z')\). Let \(Y= (y_1,\dots, y_n)= ((x_1,x'_1),\dots, (x_n,x'_n))\). (Here we get \((x_i,x'_i)\in D\) since the quadruples \(Q\) and \(Q'\) are similar.)

It follows from the definition of similarity that the pairs of the form \((m^{-1}, m')\) form an (abelian) normal subgroup \(K\) in \(D\). The elements \(m=(m,1)\) and \(m'=(1,m')\) have the same image, denoted by \(m''\) in the group \(M''=L/K\), and \((z'')^{-1}m''z'' = (z^{-1}mz)''\) for \(z''\in H''=D/L\). Denoting by \(X''\) and \(U''=\{v(z''), z''\in H''\}\) the images of \(Y\) and \(U_0\) in \(H''= D/K\), we get \(v(y'')v(z'') = f(y,z)''(f'(y,z))''v(yz)''\) and \(x''_i = (g_i)''(h_i)''v(z'')\).

These equalities show that the quadruple \(Q'' = (H'', M'', U'', X'')\) is similar to both \(Q\) and \(Q'\). We call this quadruple the sum \(Q+Q'\). Replacing the summands by isomorphic quadruples changes the sum only up to isomorphism; hence the sum operation is well defined on isomorphism classes of quadruples. If \(\cal C\) is a class of similar quadruples, we denote by \(Ext_{\cal C}\) the set of the isomorphism classes of the quadruples from \(\cal C\).

Up to isomorphism, the sum is determined by the products \(f(y,z)''(f'(y,z))''\) and \((g_i)''(h_i)''\) of elements from the abelian group \(M''\), and so the operation \(+\) is associative and commutative. The zero element of \(Ext_{\cal C}\) is the quadruple \(Q=(H,M,U,X)\), where all \(f(x,y)\) and \(g_i\) are trivial in \(M\). This means that the normal subgroup \(M\) has a semidirect complement \(S\) in \(H\) and the components of \(X\) are chosen in \(S\). Hence \(eQ=0\) for any quadruple from \(Ext_{\cal C}\) if \(M^e = 1\) for an integer \(e\). Thus, \((e-1)Q\) is the additive inverse for \(Q\), and the lemma is proved.

Up to isomorphism, every quadruple \(Q'\in\cal C\) is defined by the choice of the \(f\)- and \(g\)-elements from \(M\) in the above notation. There are fewer than \(|H|^3\) possibilities for the choice of \(f\)-elements and fewer than \(|H|^n\) possibilities for the choice of \(g\)-elements. This proves the lemma.

Note that the construction of the sum of two quadruples can be almost literally extended to the case of \(s\ge 2\) quadruples from \(Ext_{\cal C}\). Namely, let \(D\) be the inverse image in \(P=H_1\times\dots\times H_s\) of the diagonal subgroup of \(P/L\), where \(L=M_1\times\dots\times M_s\). The \(n\)-tuple \(Y\) is defined diagonally, as in the case \(s=2\). The subgroup \(K\) is generated by all elements of the form \(m^{-1}m'\), where \(m\) belongs to some subgroup \(M_k\) and \(m'\) is the copy of \(m\) in some \(M_l\). Then \(H'' = D/K\), \(M''=L/K\), and the definitions of \(U''\) and \(X''\) are similar to the case \(s=2\), with all \(s\) quadruples taken into account. Again this gives a homomorphism \(D\to H_0\), mapping the \(n\)-tuple \(Y\) componentwise to \(X_0\).

Let \(D_I\) be the projection of \(D\) to the product \(H_{i_1}\times\dots\times H_{i_t}\) The group \(D_I\) can be defined as \(D\) but using the quadruples with indices from the set \(I=\{i_1,\dots,i_t\}\) only. The image of \(Y\) in \(D_I\) is its projection \(Y_I\). As in the previous paragraph, there is a homomorphism \(D_I\to H_0\) mapping \(Y_I\) to \(X_0\), and the composition \(D\to D_I\to H_0\) maps \(Y\) to \(X_0\) componentwise.

Clearly, the groups \(P\) and \(D\) belong to the variety \(var(H_1,\dots,H_s)\), and
\(|D|<|P|= \prod_{i=1}^s |H_i|\), as required.

We now return to the finite group \(G\) with the nontrivial abelian normal subgroup \(N\), choose a set of coset representatives \(U=\{u(z)\mid z\in G/N\}\) and an arbitrary \(n\)-tuple \(X\) of elements from \(G\). Denote by \(\cal C\) the class of quadruples similar to \(Q=(G,N,U,X)\) and denote by \(Ext_Q\) the subset of \(Ext_{\cal C}\), which contains the quadruples \((H',M',U', X')\) from \(\cal C\) such that \(H'\in \mathfrak{V}=var\; G\).

The nonempty set \(Ext_Q\) is closed under the sum operation \(R=Q'+Q''\) since the first component \(H_0\) of \(R\) is defined as a homomorphic image of a subgroup of a direct product \(P= H'\times H''\). Therefore, if \(H', H''\in\mathfrak{V}\), then \(H_0\in \mathfrak{V}\) as well, which proves the lemma.

We also note:

The next estimate uses the group structure on the set \(Ext_Q\).

By Lemmas 8 and 5, the set \(Ext_Q\) is an abelian group whose order is less than \(|G|^{n+3}\) by Lemma 6. Therefore one may choose a generating set \(S\) for \(Ext_Q\) of cardinality \(r< \log_2(|G|^{n+3})\le \kappa n\), where the constant \(\kappa\) does not depend on \(n\). Take \(e\) copies of each generator from \(S\) (where \(N^e=1\) and so \(eExt_{\cal C}=0\)) and obtain \(s=er\) quadruples \(Q_j=(H_j, M_j, U_j, X_j)\), \(j=1,\dots s\). Then each quadruple \(Q' = (G',N',U',X')\in Ext_Q\) is equal to a sum \(Q_{i_1}+\dots + Q_{i_t}\) for some indices \(1\le i_1<\dots<i_t\le s\).

By Lemma 7, there exists a group \(D_Q\in \mathfrak{V}= var\; G\) and an \(n\)-tuple \(X\) in it (independent of the choice of \(Q'\)) such that there is a homomorphism \(D_Q\to G'\) mapping \(X\) to \(X'\). Since \(|D_Q| \le |G|^s< |G|^{er}\) and \(r<\kappa n\), we obtain the required estimate with \(K= e\kappa\).

Quadruples of the form \((G,N,U,X)\) may belong to different groups \(Ext_Q\) depending on the choice of the \(n\)-tuple \(X\) modulo the subgroup \(N\). Thus there are \(m^n\) different such groups, where \(m = |G/N|\).

For each group \(Ext_Q\), Lemma 9 provides us with a group \(D_Q\in \mathfrak{V}=var\; G\) and an \(n\)-tuple \(X_Q\) in it such that for every \(Q'=(H',M',U',X')\in Ext_Q\), there is a homomorphism \(D_Q\to H'\) that maps \(X_Q\) to \(X'\).

Let \(B\) be the direct product of \(m^n\) groups \(D_Q\) corresponding to different groups \(Ext_Q\), and choose the \(n\)-tuple \(X_0\) in \(B\) with projections \(X_Q\) to the direct factors \(D_Q\). Then every map from the components of \(X_0\) to \(G\) extends to a homomorphism from \(B\) to \(G\). The same is true for the subgroup \(C\) of \(B\) generated by the components of \(X_0\). Since both \(B\) and \(C\) belong to the variety \(\mathfrak{V}\), this property implies that \(C\) is an \(n\)-generated \(\mathfrak{V}\)-free group (see [2], 13.21).

Thus by Lemmas 9 and 6, \(|F_n|=|C|\le |B| < (|G|^{Kn})^{m^n}\). Taking the \(n\)th root of the logarithm of the right-hand side, we get \(m(1+o(1))\), whence \(pt(G)\le m<|G|\). This proves the lemma and completes the proof of Theorem 1.

4 Proof of Theorem 2 and questions↩︎

Note that a finite prime ring \(A\) cannot contain two different minimal ideals, and multiplication is nonzero on the unique minimal ideal. If, for brevity, rings with zero multiplication are called abelian (as in Lie theory), \(A\) is a monolithic ring with a non-abelian monolith \(N\).

Such a reformulation makes Theorem 2 very similar to Theorem 1, and the proofs are similar as well. Of course, the variety \(\mathfrak{V}=var\; A\) contains rings, not groups; one works with ideals instead of normal subgroups, and elementwise commutation is replaced by mutual annihilation. No other modifications are needed for the proof of sufficiency in Theorem 2 or for the proof of the analogue of Lemma 3.

The conjugation action of a group \(H/M\) on an abelian normal subgroup \(M\) is now replaced with two "actions": multiplication of elements \(m\) of the ideal \(M\) by elements \(x\) of the ring \(H/M\) on the left and on the right, that is, \(xm, mx\in M\). For the two ring operations, we now have two corresponding sets of equalities for coset representatives: \(u(y)+u(z) = f(y,z) + u(y+z)\) and \(u(y)u(z) = e(y,z) + u(yz)\), where \(U=\{u(z)\mid z\in H/M\}\) and \(f(y,z), e(y,z)\in M\). For two similar quadruples \(Q\) and \(Q'\), we now have isomorphisms \(M\to M'\) and \(H/M \to H'/M'\) such that \((xm)' = x'm'\) and \((mx)'=m'x'\) for any \(m\in M\) and \(x\in H/M\). If \(Q=(H,M,U,X)\) and \(Q'=(H',M',U',X')\) are similar, then for any component \(x_i\) of \(X\) the equality \(x_i=g_i+u(z)\) for \(g_i\in M\) implies \(x'_i = h_i+u'(z')\) for some \(h_i\in M'\). \(Q\) and \(Q'\) are called isomorphic if \(f'(x',y') = f(x,y)'\), \(e'(x',y') = e(x,y)'\) and \(h_i=(g_i)'\) for all \(x,y,i\).

The definition of the sum \(Q+Q'\) and Lemma 5 remain the same up to the obvious change of the vocabulary. Since we now add \(e\)-elements, the right-hand side of Lemma 6 is replaced with \(|H|^{n+6}\). To complete the proof of Theorem 2, it suffices to make the corresponding terminological changes in the proofs of Lemmas 7–10.

It is unknown whether there exists a finite group with a non-integer potential. It is also unknown whether one may use \(\lim_{n\to \infty}\) instead of \(\limsup_{n\to \infty}\) in the definition of the potential of a finite group. Similar questions for finite nonassociative algebras have been formulated in [3].

An extension of Theorem 2 to finitely generated varieties of \(\Omega\)-algebras suggests itself.

Acknowledgments. The author thanks Professor Yu. Bahturin for helpful discussions.

References↩︎