Abstract
Let p be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite p-groups whose center has index \(p^3\), which is a strong contrast to the analogous situation for \(p = 2\).
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1 Introduction
Let p be a prime number and let F be the field with p elements. In this paper, we study the modular isomorphism problem, which can be stated as follows:
Consider finite p-groups G and H such that the group algebras FG and FH are isomorphic. Is it true that G and H are isomorphic?
In the past 50 years, this problem has seen extensive research. There are many classes of finite p-groups for which it is known to have a positive answer, such as the class of abelian p-groups [5], the class of p-groups of nilpotency class 2 with elementary abelian derived subgroup [22], or the class of metacyclic groups [1, 23]. The modular isomorphism problem has a positive answer for groups with a center of index \(p^2\). This follows from the earlier-mentioned result by Sandling [22]. Moreover, the generalization of the problem for this class of groups to arbitrary fields of characteristic p has a positive solution, as shown by [6]. The modular isomorphism problem is also known to have a positive answer for groups of order dividing \(p^5\) for any prime [17, 20], for groups of order dividing \(2^8\), for groups of order dividing \(3^7\), and, except for a few groups, for groups of order dividing \(5^6\) [7, 14]. Some recent results are for example [3, 15], and a complete overview can be found in the recent survey [12].
Despite all positive results, it was shown in [9] that the general answer to this problem is negative. The smallest counterexample consists of a pair of groups of order \(2^9\) with centers of index 8. For \(p > 2\), the modular isomorphism problem is still open.
For a p-group G, we write \(\textrm{Z}(G)\) for its center, \(\Phi (G)\) for its Frattini subgroup, and \((\gamma _i(G))_{i \ge 1}\) for its lower central series. Moreover we write \(G^p=\left\langle g^p:g\in G \right\rangle \). Motivated by the counterexample of [9] for \(p = 2\), we now study finite p-groups with a center of index \(p^3\) for \(p > 2\). In contrast to the situation for \(p = 2\), it turns out that the modular isomorphism problem has a positive answer for this class of groups. Our main result is the following:
Theorem A
Let \(p>2\) be a prime integer, let F be the field with p elements, and let G and H be finite p-groups. Suppose that \(|G:\textrm{Z}(G)|=p^3\). If \(FG\cong FH\), then \(G\cong H\).
Note that the class of p-groups G with \(|G:\textrm{Z}(G)| = p^3\) is not contained in the union of the classes of p-groups for which the modular isomorphism problem is known to have a positive answer. Corresponding examples are discussed in Section 4. The key ingredient for the proof of Theorem A is the following result, which might also be of independent interest:
Theorem B
Let \(p>2\) be a prime integer, let F be the field with p elements, and consider finite p-groups G and H with \(F G\cong F H\). Let \(d \in {{\mathbb {Z}}}_{\ge 0}\) with \(|G/\Phi (G)\textrm{Z}(G)|=p^d\). If
-
(1)
\(G^p\cap \gamma _2(G)\subseteq \gamma _2(G)^p \gamma _3(G)\) and
-
(2)
\(|\gamma _2(G)/\gamma _2(G)^p\gamma _3(G)|=p^{{d\atopwithdelims ()2}},\)
then \(G/\gamma _2(G)^p \gamma _4(G)\cong H/\gamma _2(H)^p \gamma _4(H)\).
This statement is a generalization of [14, Theorem 1.2] when \(p>2\), as for \(d=2\), it is equivalent to the following implication: if \( G/\textrm{Z}(G )\) is 2-generated, then the isomorphism class of \(G/\gamma _2(G)^p\gamma _4(G)\) is determined by FG (see Corollary 3.5).
This paper is organized in the following way: In Section 2, we introduce the notation used in this paper. In Section 3, we derive results on the so-called small group algebra, which will lead to the proof of Theorem B. Finally, in Section 4, we prove Theorem A.
2 Preliminaries and notation
Let p be an odd prime number and let G be a finite p-group. We use the standard group-theoretic notation. In particular, as above, Z(G) and \(\Phi (G)\) denote the center and the Frattini subgroup of G, respectively. We set \(G^p = \langle g^p :g \in G \rangle \). A Burnside basis is a generating system of G of minimal size. For \(a,b \in G\), let \([a,b] = a^{-1} b^{-1} a b\). Moreover, we set \(\gamma _1(G) = G\) and \(\gamma _{i+1}(G) = [\gamma _i(G), G]\) for \(i\ge 1\).
Lemma 2.1
Let G be a finite p-group of nilpotency class at most 3. Then for each \(x,y\in G\),
Proof
Using several times the identity
we derive that
A similar argument shows the second identity. \(\square \)
As an immediate consequence, we obtain:
Lemma 2.2
Let G be a finite p-group with \(p>2\). If \(\gamma _2(G)^p\gamma _4(G)=1\), then \(G^p\subseteq \textrm{Z}(G)\).
Let F be a field of characteristic p. In this paper, we only consider unitary, finite-dimensional F-algebras. An augmented F-algebra is an F-algebra A endowed with an F-algebra homomorphism \(A\rightarrow F\), called the augmentation map. The group algebra FG is an augmented F-algebra with augmentation map \(\sum _{g\in G} a_gg\mapsto \sum _{g\in G} a_g\). The kernel I(A) of the augmentation map is called the augmentation ideal of A. For \(A=FG\), we abbreviate I(FG) by I(G).
Let A be an augmented local F-algebra, as it is the case for \(A=FG\). Then we denote \(V(A)= 1+I(A)\), the subgroup of the group of units of A formed by the elements of augmentation 1. Every quotient of A by some ideal J of A is also an augmented algebra with an augmentation map induced by the augmentation of A. We can (and will) make the identifications \(I(A/J)=I(A)/J\) and \(V(A/J)=1+I(A/J)=V(A)/(1+J)\). We define the set K(A) to be the F-subspace of A generated by the elements of the form \(ab-ba\) with \(a,b\in A\).
Note that \(I(G) I(\gamma _2(G))\) is a 2-sided ideal of FG. A main tool in this paper will be the investigation of the small group algebra \(S = FG/I(G) I(\gamma _2(G))\). This object has been extensively studied by R. Sandling and M.A.M. Salim (see [18, 19, 22]). It was used in [20] to positively answer the modular isomorphism problem for groups of order dividing \(p^5\). Moreover, the analogous construction for group rings over the integers was used in A. Whitcomb’s thesis [26] to positively answer the isomorphism problem for integral group rings for metabelian groups. Further results about the small group algebra in a wider sense can be found in [10]. By a bar, we denote both the natural projection \({\overline{\cdot }}:FG\rightarrow S\), which is an F-algebra homomorphism, and its restriction \({\overline{\cdot }}:V(FG)\rightarrow V(S)\), which can be viewed as a group homomorphism. A subgroup of V(FG) that is an F-basis for FG is called a group basis of FG. We say a property P is determined by the group algebra if for any pair of p-groups G and H such that \(FG\cong FH\), the group G satisfies P if and only if H satisfies P.
We have that \((1+I(G)I(\gamma _2(G)))\cap G=\Phi (\gamma _2(G))\) (see the introduction of [19]), so \(\overline{G} \cong G/\Phi (\gamma _2(G))\). Note that this holds for any group basis of FG. However, it has been known for a long time that the information encoded by the small group algebra alone is not enough to recover the isomorphism type of \(G/\Phi (\gamma _2(G))\) (see [2, Section 4]), even for groups of nilpotency class 3 (see [15, Example 3.11] and the subsequent discussion).
We frequently use the following result due to Sandling:
Lemma 2.3
([22], see also [14, Theorem 3.2]). Let G be a finite p-group. Let \(g_1, \ldots , g_m \in \overline{G}\) be elements whose images in \(\overline{G}/\gamma _2(\overline{G})\) form an independent generating set as an abelian group. Let A be the subgroup of V(S) generated by the elements \(1 + (g_1-1)^{k_1} \cdots (g_m-1)^{k_m}\) with \(k_1, \ldots , k_m \in {{\mathbb {Z}}}_{\ge 0}\) and \(k_1 + \dots + k_m \ge 2\). Then the following hold:
-
(1)
\(V(S) = \overline{G}A\).
-
(2)
For \(a = 1 + (g_1-1)^{k_1} \cdots (g_m-1)^{k_m} \in A\) and \(g \in \overline{G}\), we have
$$\begin{aligned} \begin{aligned} \left[ g,a\right] = [g, g_1, g_1, \dots , g_1, g_2, \dots , g_2, \dots , g_m, \dots , g_m],\end{aligned} \end{aligned}$$where each \(g_i\) appears exactly \(k_i\) times.
-
(3)
If additionally \(\gamma _4(G) = 1\) holds, then A is abelian and \(V(S) = \overline{G} \rtimes A\).
Furthermore, [19, Theorem 1.1] yields that
for each \(i\ge 2\).
3 Results on the small group algebra
Throughout, let F denote the field with p elements for a prime number \(p> 2\) and let G be a finite p-group. In this section, we derive results on the small group algebra that will later be used to solve the modular isomorphism problem for groups with a center of index \(p^3\).
We use the notation introduced in the preceding section. In particular, let \(\overline{\cdot } :FG\rightarrow S\) be the natural projection onto the small group algebra S and let \(V(S) = 1 + I(G)/I(G) I(\gamma _2(G))\) denote the group of units of augmentation 1 of S. Furthermore, let A be a subgroup of V(S) as in Lemma 2.3, so \(V(S)={\overline{G}} A \).
Remark 3.1
For every normal subgroup N of G contained in \(\gamma _2(G)\), it follows easily from the identity \(xy-1=x-1+y-1+(x-1)(y-1)\) that
In particular, \(\gamma _i({\overline{G}})-1 = \overline{\gamma _i(G)-1}\) is an ideal in S. In the following, we set \(\Gamma = \gamma _4(G) -1 \subseteq FG\) and consider the quotient algebra \(S_0 = FG/(I(G) I(\gamma _2(G)) + \Gamma ) \cong S/\overline{\Gamma }\) together with the corresponding projection \(\pi _0:FG \rightarrow S_0\). Then \(\pi _0(G)= G/\gamma _2(G)^p \gamma _4(G)\), and the same holds for every group basis of FG. Moreover, \(S_0\) is isomorphic to the small group algebra of \(\pi _0(G)\). Let \(A_0\) be the image of A in \(V(S_0)\). Then by Lemma 2.3, we have that \(V(S_0)= \pi _0(G) \rtimes A_0\). More generally, the following holds:
Lemma 3.2
([15, Proposition 3.1]). Let H be any group basis of FG. Then \(V(S_0)=\pi _0(H)\rtimes A_0\).
Finally, we consider the natural projection from \(V(S_0)\) onto \(V= V(S_0)/ (A_0\cap \textrm{Z}(V(S_0)))\). Composing it with the restriction of \(\pi _0\) to V(FG) yields a group homomorphism
Hence \(V=\pi (G)\rtimes E\) follows, where \(E=A_0/ (A_0\cap \textrm{Z}(V(S_0)))\) is elementary abelian due to \(A_0^p\subseteq \textrm{Z}(V(S_0))\) (see (2.1) and Lemma 2.2). Therefore, by Lemma 3.2, we have
for any group basis H of FG.
In the next lemma, we derive a presentation for groups satisfying the hypotheses of Theorem B.
Lemma 3.3
Write \(|G/\textrm{Z}(G)\Phi (G)|=p^d\), \(|\gamma _2(G)^p \gamma _3(G)/\gamma _2(G)^p\gamma _4(G)|=p^m\), and \(|G/\Phi (G)|=p^k\). If
-
(a)
\(G^p\cap \gamma _2(G)\subseteq \gamma _2(G)^p \gamma _3(G)\) and
-
(b)
\(|\gamma _2(G)/\gamma _2(G)^p\gamma _3(G)|=p^{{d\atopwithdelims ()2}} \),
then there exist lists of integers
with \(0<n_i\) and \(0\le \alpha _{is}, \beta _{ijls}\le p-1\) such that
where \({\mathcal {R}}_{( {\varvec{n}},\varvec{\alpha },\varvec{\beta } )} ( g_1,\dots , g_k,c_1,\dots , c_m) \) is the set formed by the following relations:
-
1.
\(g_i^{p^{n_i}}=\prod _{1\le s \le m}c_s^{\alpha _{is}}\) for each \(1\le i \le k\).
-
2.
\( [[g_i,g_j], g_l]=\prod _{1\le s\le m} c_s^{\beta _{ijls}}\) for \(1\le i,j,l\le d\).
-
3.
\([g_i,g_j]^p=1\) for \(1\le i,j\le d\).
-
4.
\(g_i\) is central for \(d<i\le k\).
-
5.
The nilpotency class is at most three (i.e., for \(t\ge 4\), all t-fold commutators of the generators vanish).
Moreover, the isomorphism \(\phi :G/\gamma _2(G)^p\gamma _4(G)\rightarrow \left\langle g_1,\dots , g_k,c_1,\dots , c_m \right\rangle \) can be chosen such that the elements \(\phi ^{-1}(g_{d+1})\),..., \(\phi ^{-1}(g_k)\in G/\gamma _2(G)^p \gamma _4(G)\) have central preimages in G.
Proof
Set \({\widetilde{G}}=G/\gamma _2(G)^p \gamma _4(G)\) and, for \(g \in G\), write \({\widetilde{g}} = g \gamma _2(G)^p \gamma _4(G) \in {\widetilde{G}}\) (similarly for subsets of G). Observe that \(|\gamma _3({\widetilde{G}})|=p^m\) and \(|{\widetilde{G}}/\Phi ({\widetilde{G}})|=p^k\). Write \( |{\widetilde{G}}/\textrm{Z}({\widetilde{G}}) \Phi ({\widetilde{G}})|=p^t\). As the image of \(Z(G) \Phi (G)\) in \({\widetilde{G}}\) has index at most \(p^d\), we obtain \(p^t\le p^d\). Moreover, \(\gamma _2({{\widetilde{G}}})\) is elementary abelian and the hypotheses imply that \( {\widetilde{G}}^p \cap \gamma _2({{\widetilde{G}}})\subseteq \gamma _3({{\widetilde{G}}}) \) and \(|\gamma _2({{\widetilde{G}}})/\gamma _3({{\widetilde{G}}})|=p^{d\atopwithdelims ()2}\). Since \({{{\widetilde{G}}}}^p\subseteq \textrm{Z}({\widetilde{G}})\) by Lemma 2.2, we have a well-defined map \({\mathfrak {c}} :{\widetilde{G}}/\textrm{Z}({\widetilde{G}}) \Phi ({\widetilde{G}}) \times {\widetilde{G}}/\textrm{Z}({\widetilde{G}}) \Phi ({\widetilde{G}}) \rightarrow \gamma _2({{\widetilde{G}}})/\gamma _3({{\widetilde{G}}}) \) induced by the group commutator map, which is bilinear and anti-symmetric. If \(\{x_1,\dots , x_t\}\) is a basis of \({\widetilde{G}}/\textrm{Z}({\widetilde{G}}) \Phi ({\widetilde{G}})\), then \({\mathfrak {c}}(x_i,x_j)\) with \(1\le i <j \le t\) generates \(\gamma _2({{\widetilde{G}}})/\gamma _3({{\widetilde{G}}})\) as an F-vector space. Thus \(p^{{d\atopwithdelims ()2}}=|\gamma _2(\widetilde{G})/\gamma _3({{\widetilde{G}}})| \le p^{{t\atopwithdelims ()2}}\). This yields \(t=d\).
Let \(\{x_1,\dots , x_d\}\) be a set of preimages of a basis of \( G/\textrm{Z}( G)\Phi ( G)\) in G, and let \(\{x_{d+1},\dots x_k\)} be a set of central preimages in G of a basis of \(\textrm{Z}(G)\Phi (G)/\Phi (G)\). Thus \(\{x_1,\dots , x_k\}\) is a Burnside basis of G, and its image is a Burnside basis of \({{\widetilde{G}}}\). By abuse of notation, we regard \(\{x_1,\dots , x_k\}\) as a subset of \({{\widetilde{G}}}\). Let \(\{z_1,\dots , z_m\}\) be a basis of \(\gamma _3({{\widetilde{G}}})\). Write \(p^{n_i}\) for the order of \(x_i \gamma _2({\widetilde{G}})\) in \({\widetilde{G}}/\gamma _2({{\widetilde{G}}})\). As \({\widetilde{G}}^p\cap \gamma _2({{\widetilde{G}}}) \subseteq \gamma _3({{\widetilde{G}}})\), we have that \(x_i^{p^{n_i}}=\prod _{1\le s \le m} z_s^{\alpha _{i s}}\) for uniquely determined \(\alpha _{is} \in \{0, \dots , p-1\}\). Moreover, for \(1\le i,j,l\le d\), we can write \( [[x_i,x_j], x_l]=\prod _{1\le s\le m} z_s^{\beta _{ijls}}\) for uniquely determined \(\beta _{ijls} \in \{0, \dots , p-1\}\). As \(\{z_1, \ldots , z_m\}\) is a basis of \(\gamma _3({{\widetilde{G}}})\), the matrix \((\beta _{ijls})_{ (i,j,l), s}\), with rows indexed by triples (i, j, l) and columns indexed by s, has rank m.
Define \({\varvec{n}}, \varvec{\alpha }\), and \(\varvec{\beta } \) as in (3.3) and let
The assignment \(g_i \mapsto x_i\) (\(i = 1, \ldots , k\)) and \(c_j \mapsto z_j\) (\(j = 1, \ldots , m\)) defines a surjective group homomorphism \(H\rightarrow {{\widetilde{G}}}\). To complete the proof, it suffices to show that it is an isomorphism, or equivalently that \(|H|\le |{{\widetilde{G}}}|\).
Observe that since the matrix \((\beta _{ijls})_{ (i,j,l),s}\) has rank m, relation (2) implies that \(c_1, \ldots , c_m \in \gamma _3(H)\). In particular, we have \(c_1, \ldots , c_m \in \Phi (H)\) and hence \(|H/\Phi (H)|\le p^k\).
As H has nilpotency class at most 3, \(\gamma _2(H)\) is abelian. The group \(\gamma _2(H)\) is generated by \(\gamma _3(H)\) and elements of order p by relation (3), and hence \(\gamma _2(H)/\gamma _3(H)\) is elementary abelian. Thus so is \(\gamma _2(H)/\gamma _2(H)\cap \textrm{Z}(H)\). On the other hand, the image of the map \(H/\textrm{Z}(H)\times \gamma _2(H)/\gamma _2(H) \cap \textrm{Z}(H)\rightarrow \gamma _3(H)\) induced by the commutator map generates \(\gamma _3(H)\). For \(h \in H\) and \(h' \in \gamma _2(H)\), we have \([h \textrm{Z}(H), h' \gamma _2(H)\cap \textrm{Z}(H)]^p= [h\textrm{Z}(H), (h')^p \gamma _2(H)\cap \textrm{Z}(H)]=1\) (see Lemma 2.1). Thus \(\gamma _3(H)\) is elementary abelian. This implies that \(\gamma _2(H)\) is elementary abelian (as an abelian group generated by elements of order p).
By relations (2) and (5), we have \(\gamma _3(H) = \langle c_1, \dots , c_m\rangle \), so
Moreover, using (4), we derive that \(|H/\textrm{Z}(H)\Phi (H)|\le p^d\). Observe that \([H,\Phi (H)] \subseteq \gamma _3(H)\) due to \([x^p,y] \gamma _3(H) = [x,y]^p \gamma _3(H)\) for all \(x,y \in H\) (see Lemma 2.1) and \(\gamma _2(H)\) being elementary abelian. Then the commutator in the group defines a bilinear map \(H/Z(H) \Phi (H) \times H/Z(H) \Phi (H)\rightarrow \gamma _2(H)/\gamma _3(H)\), \((x\textrm{Z}(H)\Phi (H), y\textrm{Z}(H)\Phi (H))\mapsto [x,y] \gamma _3(H)\). As for \({\widetilde{G}}\), we derive that
Finally, \(H/\gamma _2(H)=\left\langle g_1 \gamma _2(H),\dots , g_k \gamma _2(H) \right\rangle \) has order at most \(\prod _{i=1 }^k p^{n_i}=|\widetilde{G}/\gamma _2({{\widetilde{G}}})|\) by relation (1). Since \( |{{\widetilde{G}}}|= |{{\widetilde{G}}}/\gamma _2({{\widetilde{G}}})|\cdot |\gamma _2({{\widetilde{G}}})/\gamma _3({{\widetilde{G}}}) |\cdot |\gamma _3({{\widetilde{G}}})| \), combining this with (3.4) and (3.5) yields that \(|H|\le |\widetilde{G}|\). This proves the statement. \(\square \)
Theorem 3.4
Let p be an odd prime number and let F be the field with p elements. Consider finite p-groups G and H with \(FG \cong FH\) and let \(|G/\Phi (G)\textrm{Z}(G)|=p^d\). If
-
(a)
\(G^p\cap \gamma _2(G)\subseteq \gamma _2(G)^p \gamma _3(G)\) and
-
(b)
\(|\gamma _2(G)/\gamma _2(G)^p\gamma _3(G)|=p^{{d\atopwithdelims ()2}},\)
then \(G/\gamma _2(G)^p \gamma _4(G)\cong H/\gamma _2(H)^p \gamma _4(H)\).
Proof
Consider the isomorphism of elementary abelian p-groups
(see [24, Propositions III.1.14 and III.1.15]). We have an F-vector space decomposition \(\textrm{Z}(FG) = F\textrm{Z}(G) + \textrm{Z}(FG) \cap K(FG)\) (see [21, Lemma 6.10]). It is well-known that \(FG \cdot K(FG) = K(FG) \cdot FG = I(\gamma _2(G)) FG\). As \(\gamma _2(G) \subseteq \Phi (G) = (1 + I(G)^2) \cap G\) holds (see [11, Theorem 5.5]), we have
In particular, \(\varphi \) restricts to an isomorphism
Of course, the analogous reasoning applies to H. In particular, we obtain
Moreover, note that the conditions (a) and (b) depend only on the quotient \(G/\gamma _2(G)^p \gamma _3(G)\), which is determined by FG by [22]. Thus, if G satisfies (a) and (b), then the analogous statements hold for H.
Consider the group V and the projection \(\pi :V(FG) \rightarrow V\) introduced in (3.1). Let \(\{g_1,g_2,\dots , g_k, c_1,\dots ,c_m\}\) be a set of generators of \(\pi (G)\) satisfying relations
as in Lemma 3.3. For \(1\le i \le k\), the decomposition \(V=\pi (H)\rtimes E\) (see (3.2)) guarantees that \(g_i=h_i e_i\) for some unique \(h_i\in \pi (H)\) and \(e_i\in E\). Moreover \(c_j \in \gamma _2(\pi (G))=\gamma _2(S)=\gamma _2(\pi (H))\) for \(j=1,\dots , m\) (see (2.1)). Now let \(d<i \le k\), so \(g_i \in Z(\pi (G))\). We have
using \(e_i -1 \in I(G)^2\) (see Lemma 2.3). In particular, this yields \(h_i -1 + I(G)^2 \in (Z(FG) \cap I(G) + I(G)^2)/I(G)^2\). Using the isomorphism in (3.7), and the analogue for H, we derive that \(h_i ={\hat{h}}_i {{\widetilde{h}}}_i\), with \({\hat{h}}_i\in \textrm{Z}(\pi (H)) \) and \({{\widetilde{h}}}_i\in \Phi (\pi ( H)) = \gamma _2(\pi (H)) \pi (H)^p\). Observe that \(\pi (H)^p\) is central in \(\pi (H)\) by Lemma 2.2, so we can furthermore assume that \({{\widetilde{h}}}_i \in \gamma _2(\pi (H))\). Then \(\{h_1,\dots , h_d, {\hat{h}}_{d+1}, \dots , {\hat{h}}_k,c_1,\dots , c_m\}\) forms a generating set of \(\pi (H)\). We now show that it satisfies the relations
Then H is an epimorphic image of G of the same size and we obtain \(G\cong H\) as desired.
We check the relations one by one. Relations (3), (4), and (5) are immediate. For \(1\le i \le k\), one has
and, in particular, for \(d<i\le k\), one has \(g_i^p=h_i^p= {\hat{h}}_i^p {{\widetilde{h}}}_i^p = {\hat{h}}_i ^p\). Then condition (1) follows. Condition (2) also follows readily because, for \(1\le i \le d\) and arbitrary \(x_i,x_j,x_l\in A\), we have that
This completes the proof. \(\square \)
The next result shows that Theorem 3.4 is a generalization of [14, Theorem 1.2].
Corollary 3.5
Let p be an odd prime number, let F be the field with p elements, and let G and H be finite p-groups such that \(F G\cong F H\). If \(|G/\Phi (G)\textrm{Z}(G)|=p^2\), then
Proof
If \(G/ \gamma _2(G)^p \gamma _4(G)\) has nilpotency class at most 2, then so does \(H/\gamma _2(H)^p \gamma _4(H)\) (see (2.1)) and hence the result follows by [22]. Thus we assume that G has class 3. Observe that \(\gamma _2(G)/\gamma _2(G)^p\gamma _3(G)\) has order p, as it is generated by \([g_1,g_2]\gamma _2(G)^p\gamma _3(G)\) if we consider a set of generators \(\{g_1,\dots ,g_k,c_1,\dots , c_m\}\) as in Lemma 3.3. Observe that every p-power is central in \(G/ \gamma _2(G)^p \gamma _4(G)\) by Lemma 2.2. On the other hand, the image of \([g_1,g_2]\) in \(G/\gamma _2(G)^p \gamma _4(G)\) is not central as G does not have nilpotency class 2. This yields that \(G^p \cap \gamma _2(G)\subseteq \gamma _2(G)^p \gamma _3(G)\). Thus the hypotheses of Theorem 3.4 with \(d=2\) hold, and the statement follows. \(\square \)
4 Groups with a center of index \(p^3\)
Let p be a prime number. In this section, we use the results about the small group algebra from Section 3 to give a positive answer to the modular isomorphism problem for groups with center of odd index \(p^3\).
Lemma 4.1
Let p be an odd prime number and let G be a finite p-group with \(|G:\textrm{Z}(G)|=p^3\). Then:
-
(1)
The nilpotency class of G is at most 3.
-
(2)
\(\gamma _2(G)\) is elementary abelian.
-
(3)
Either G has nilpotency class 2 or \(|G/\Phi (G)\textrm{Z}(G)|=p^2\).
Proof
Examination of the upper central series yields that the nilpotency class of G is at most 3. The quotient \(G/\textrm{Z}(G)\) is isomorphic to one of the five groups of order \(p^3\), namely to one of \(C_{p^3}\), \(C_p^3\), \(C_{p^2}\times C_p\), \((C_p\times C_p)\rtimes C_p\), or \(C_{p^2}\rtimes C_p\). Observe that G/Z(G) is not cyclic as G is abelian otherwise.
Next, we show that \(\gamma _2(G)\) is elementary abelian. It is clear that \(\gamma _2(G)\) is abelian as \([\gamma _2(G), \gamma _2(G)]\subseteq \gamma _4(G)=1\). Thus it suffices to show that the generators of \(\gamma _2(G)\) have order p. Note that \(\gamma _3(G)\) is elementary abelian. Indeed, it is generated by elements of the form [x, y] with \(x\in G\) and \(y\in \gamma _2(G)\); since \(\gamma _2(G)\textrm{Z}(G)/\textrm{Z}(G)\) has order at most p, we have \(y^p\in \textrm{Z}(G)\), so Lemma 2.1 shows that \(1=[x,y^p]=[x,y]^p \). Let \(a,b_1, b_2\in G\) be such that \(G/\textrm{Z}(G)=\left\langle a\textrm{Z}(G), b_1 \textrm{Z}(G), b_2\textrm{Z}(G) \right\rangle \). By the structure of \(G/\textrm{Z}(G)\), we may assume that \( b_i^p\in \textrm{Z}(G) \) for \(i=1,2\). Then for every \(x\in G\) and \(i \in \{1,2\}\), Lemma 2.1 yields that
This shows that \(\gamma _2(G)\) is elementary abelian.
Finally, if G has nilpotency class 3, then \(G/\textrm{Z}(G)\) is isomorphic to one of the two non-abelian p-groups of order \(p^3\). Then it is clear that \(|\gamma _2(G)\textrm{Z}(G)/\textrm{Z}(G)|=p\), and hence \(|G/\Phi (G)\textrm{Z}(G)|=p^2\). \(\square \)
We can now prove our main result:
Theorem 4.2
Let p be an odd prime number, let F be the field with p elements, and let G and H be finite p-groups. Suppose that \(|G:\textrm{Z}(G)|=p^3\). If \(FG\cong FH\), then \(G\cong H\).
Proof
By Lemma 4.1, we know that \(\gamma _2(G)\) is elementary abelian. If the nilpotency class of G is 2, then \(G\! \cong \! H/\gamma _2(H)^p \! \gamma _3(H)\) by [22], so \(G \! \cong \! H\) due to \(|G|\! =\! |H|\). Thus, by Lemma 4.1, we can assume that the nilpotency class of G is 3, and that \(|G/\Phi (G)\textrm{Z}(G)|\! =\! p^2\). By Corollary 3.5, we then have \(G\cong H\). \(\square \)
Remark 4.3
We point out that there cannot be an analogue of Theorem 4.2 for \(p=2\), as in [9] non-isomorphic finite 2-groups with centers of index 8 and isomorphic group algebras over every field of characteristic 2 are presented. Hence our result underlines the difference between the cases \(p=2\) and \(p>2\) for this problem (we refer to [8] for other contrasts).
Remark 4.4
Observe that [4, Corollary 8.2] yields that the finite p-groups (p odd) with extraspecial central quotient are exactly the groups with center of index \(p^3\) and nilpotency class 3. Thus Theorem 4.2 gives a positive answer to the modular isomorphism problem for this class of groups.
We conclude this section with examples illustrating that the class of p-groups with a center of index \(p^3\) is not contained in one of the classes for which the modular isomorphism problem was previously known to have a positive answer.
Example 4.5
-
(1)
Consider the following groups of order \(5^7\):
$$\begin{aligned}&G=\left\langle a,b,c,d \left| \begin{array}{c} a^{625} = b^5=c^{25}=[b,a]^5=[c,a]=[c,b] =[d,a]=[d,b] =[d,c] =1, \\ a^5=d, \left[ [b,a], a \right] = d^5, \left[ [b,a],b\right] = c^5 \end{array} \right. \right\rangle , \\&H= \left\langle a,b,c,d \left| \begin{array}{c} a^{625} = b^5=c^{25}=[b,a]^5=[c,a]=[c,b] =[d,a]=[d,b] =[d,c] =1, \\ a^5=d, \left[ [b,a], a \right] = c^5, \left[ [b,a],b\right] = d^5 \end{array} \right. \right\rangle , \\&K= \left\langle a,b,c,d \left| \begin{array}{c} a^{625} = b^5=c^{25}=[b,a]^5=[c,a]=[c,b] =[d,a]=[d,b] =[d,c] =1, \\ a^5=d^2, \left[ [b,a], a \right] = c^5, \left[ [b,a],b\right] = d^5 \end{array} \right. \right\rangle . \end{aligned}$$In GAP, they are listed as \(\texttt {SmallGroup}(5^7, 1599)\), \(\texttt {SmallGroup}(5^7,1734)\), and \(\texttt{SmallGroup}(5^7,1766)\), respectively. Each of these groups is 3-generated with minimal set of generators \(\{a,b,c\}\). The center is given by \( \langle c,d \rangle \cong C_{25} \times C_{25}\), and hence it has index \(5^3\), so Theorem A applies to G, H, and K. On the other hand, a quick verification shows that none of these groups is covered by any of the cases for which MIP is known (according to the survey in [12]). Moreover, these three groups agree in all the group theoretical invariants implemented in the GAP package ModIsomExt [13, 25].
-
(2)
Let \(p \ge 7\). In the list of groups of order \(p^6\) given in [16], consider the groups \(G_{6,3}\) as well as \(G_{6,5r}\) for \(r \in \{1, \nu \}\). These groups have centers of index \(p^3\) and do not lie in the union of classes of p-groups for which the modular isomorphism problem is known to have a positive answer (according to [12]). For small values of p, one can verify computationally that these groups agree in all group theoretical invariants implemented in ModIsomExt.
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Acknowledgements
We are grateful to Leo Margolis and Ángel del Río for their comments on an earlier version of this paper, and to Taro Sakurai for bringing the reference [4] to our attention. Moreover, we thank the referees for their helpful comments.
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Brenner, S., García-Lucas, D. On the modular isomorphism problem for groups with center of index at most \(p^3\). Arch. Math. 122, 463–474 (2024). https://doi.org/10.1007/s00013-024-01977-z
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DOI: https://doi.org/10.1007/s00013-024-01977-z