The modular isomorphism problem and abelian direct factors

Let $p$ be a prime and let $G$ be a finite $p$-group. We show that the isomorphism type of the maximal abelian direct factor of $G$, as well as the isomorphism type of the group algebra over $\mathbb F_p$ of the non-abelian remaining direct factor, are determined by $\mathbb F_p G$, generalizing the main result in arXiv:2110.10025 over the prime field. In order to do this, we address the problem of finding characteristic subgroups of $G$ such that their relative augmentation ideals depend only on the $k$-algebra structure of $kG$, where $k$ is any field of characteristic $p$, and relate it to the modular isomorphism problem, reproving and extending some known results.


Introduction
Let k be a field of characteristic p, and G and H finite p-groups.The modular isomorphism problem (MIP) asks whether the existence of an isomorphism of k-algebras kG ∼ = kH implies the existence of an isomorphism of groups G ∼ = H.The most classical version of this question also assumes that k = F p , the field with p elements.Indeed, it is already mentioned by Brauer in [Bra63] as a possibly much easier particular case of the general isomorphism problem for group rings ([Bra63, Problem 2]).Despite Brauer's optimistic observation, only partial positive results for MIP have been obtained under quite severe restrictions on the structure of G and H.For example, MIP is known to have positive answer if the groups are abelian (Deskins [Des56]), metacyclic (Bagiński [Bag88], and Sandling [San96]), or have class 2 and elementary abelian derived subgroup (Sandling [San89]).Some more recent positive results and approaches can be found in [Sak20,MM22,BdR21,MS22], and an up to date state of the art, in [Mar22].The modular isomorphism problem is now known to have negative answer in general, as it is shown in [GLMdR22] that there exist non-isomorphic finite 2-groups with isomorphic group algebras over every field of characteristic 2. This counterexample makes it even more interesting to investigate which properties (weaker than the isomorphism type) of G and H must necessarily coincide when kG ∼ = kH.In any case, the original problem remains open for p an odd prime.
On the other hand, and despite all the attention MIP received, an approach that surprisingly seems to not have been exploited until very recently is to reduce the problem from the vast class of all finite p-groups to some smaller (but maybe as vast and complicated) subclass of groups.In [MSS21] it is shown that the modular isomorphism problem can be reduced to the same problem over groups without elementary abelian direct factor.We generalize this result by dropping the 'elementary' hypothesis, i.e., showing that MIP can be reduced to the problem over groups without abelian direct factors, with no restrictions on the exponent.We formalize this as follows: For a finite p-group G, consider a decomposition G = Ab(G) ⊕ NAb(G), where Ab(G) and NAb(G) are subgroups of G such that Ab(G) is abelian, and maximal such that a decomposition like that exists.From the Krull-Remak-Schmidt theorem it follows that the isomorphism types of Ab(G) and NAb(G) do not depend on the chosen decomposition, so they are group-theoretical invariants of G. Our main theorem states that we can disregard the direct factor Ab(G) in the study of MIP.Formally: As an immediate corollary, we can extend non-trivially some of the classes of groups for which MIP is known to have a positive answer.We summarize some of them in the following corollary.Lemma 4.9 seems to suggest that this problem is not completely hopeless; however the remainder of techniques used in Section 4 rely heavily on the fact that the direct factors we disregard are abelian (and hence contained in the center of the group), so new ideas and techniques would be needed to follow this path.
The paper is organized as follows.In Section 2 we set notation and present some well-known results related to the modular isomorphism problem.In Section 3 we address the problem of finding normal subgroups N of G such that their relative augmentation ideals depend only on the algebra structure of kG, and relate this problem to the MIP.Finally, in Section 4 we prove, with the help of the results in the previous section, both Proposition C and Theorem A.

Notation and preliminaries
Throughout the paper, p will denote a prime number, k a field of characteristic p, F p the field with p elements and G and H finite p-groups.The group algebra of G over k is denoted by kG and its augmentation ideal is denoted by I(G).It is a classical result that I(G) is also the Jacobson ideal of kG.For every normal subgroup N of G, we write I(N ; G) for the relative augmentation ideal I(N )kG, i.e., the two-sided ideal generated by the elements of the form n − 1 with n ∈ N .It is well-known that this ideal is just the kernel of the natural projection kG → k(G/N ), i.e., the homomorphism of algebras extending the natural projection G → G/N .Moreover, I(N ; G) ∩ (G − 1) = N − 1.For these basic facts about the augmentation ideals we refer to [Pas77, Section 1.1].We denote by [kG, kG] the vector subspace of kG generated by the elements of the form xy − yx, with x, y ∈ kG.Given a subspace U of a vector space V over k, we write codim V (U ) = dim(V ) − dim(U ) to denote the codimension of U in V .
Our group theoretic notation is mostly standard.We write ⊕ both for internal and external direct products of groups, and also for the direct sum of vector spaces.For n ≥ 1, we denote by C n the cyclic group of order n.Given n, r ≥ 1, a homocyclic group of exponent n and rank r is a group isomorphic to r times).We let |G| denote the order of the finite p-group G, Z(G) its center, {γ i (G)} i≥1 its lower central series, G = γ 2 (G) its commutator subgroup, Φ(G) its Frattini subgroup and d(G) = min{|X| : X ⊆ G and G = X } its minimum number of generators.It is well known that G/Φ(G) is an elementary abelian p-group, and d(G) = d(G/Φ(G)).If A is a homocyclic p-group, then d(A) equals the rank of A. If moreover A is elementary abelian, it can be seen as a vector space over F p with dimension d(A).A Burnside basis of G is a minimal set of generators of G, i.e., a subset of G such that its image in G/Φ(G) is a basis as vector space.We define the omega series (Ω n (G)) n≥0 and the agemo series ( n (G)) n≥0 of G by: If N is a normal subgroup of G and n ≥ 1, we also write A property of these series that we will use is that if G is abelian, then the orders of the terms completely determine the isomorphism type of G.For more on the Jennings series, see for instance [Pas77, Section 11.1] and [Seh78, Section III.1].
The next proposition collects some well-known results about MIP that will have some relevance for our results; a complete list can be consulted in [Mar22].
Proposition 2.1.Let k = F p be the field with p elements, and let G and H be finite p-groups.Suppose that G satisfies at least one of the following conditions: (1) G is abelian [Des56].
Let φ : kG → kH be an isomorphism preserving the augmentation.Then the ideal of kG generated by [kG, kG] This is a classical result that already appears in [Col62], and plays a role in the proofs of a great part of the known results about MIP (e.g., the ones involving the small group algebra such as [San89]).In Section 3 we will be interested in subgroups satisfying the property in (2.1).More concretely, given a family F of groups, we say that a map N * : F → C, where C is the class of all groups, is a subgroup assignation if given a group G in F it returns a normal subgroup N G of G. Partially following [Sal93, Section 3.1.2],we say that the subgroup assignation for G, H ∈ F and every isomorphism φ : kG → kH preserving the augmentation.Clearly if a subgroup assignation is k-canonical, then the image subgroup is a characteristic subgroup of the original group.Unless stated otherwise, in the rest of this paper all the k-canonical assignations will be over the family F of all the finite p-groups.In the next section we provide specific ways to obtain group-theoretical invariants of G determined by kG from k-canonical assignations, and also to obtain new k-canonical assignations from the known ones.

The k-canonical subgroup problem and a transfer lemma
The following general fact is widely known, as it is mentioned in the proof of [San85, Lemma 6.10].A particular case of it is stated and reproved in [MSS21, Lemma 2.7] restricted to L = Z(G) and N = t (G)G for any positive integer t, though their proof works in general.All the items in the following lemma seem to be known, at least for some specific k-canonical subgroup assignation (mostly G → G ), but to the best of our knowledge it had never been stated in general.
Lemma 3.2.Let k be a field of characteristic p, let G and H be finite p-groups and let L G and L H be normal subgroups of G and H, respectively.Assume that there is an isomorphism φ : kG → kH such that although the proof also works for an arbitrary field k of characteristic p. See also the proof of [Bag88, Lemma 2] for N = G . ( 2) is an immediate consequence of (1), since the isomorphism type of an abelian group is determined by the orders of the terms of its Jennings series.As for (3), let φ : kG → kH an isomorphism of k-algebras.Then 2) because both groups are abelian, as Γ ⊆ N Γ .For the last isomorphism in (3), observe that φ also induces an isomorphism φ : for each i ≥ 1.Hence, as the groups L G N G /N G and L H N H /N H are abelian, they must be isomorphic.Now (4) follows immediately using (2.1).
The underlying idea behind (5) is once again the proof of [San85, Lemma 6.10], which states the same result for L Γ = Γ ; similar ideas are used in [MSS21] to prove the result specialized to L G = t (G)G , for any positive integer t.It is well-known (see [San85, Lemma 6.10]) that Since φ : kG → kH maps [kG, kG] ∩ Z(kG) to [kH, kH] ∩ Z(kH), we deduce that the restriction of φ to Z(kG) ∩ I(G) induces an isomorphism φ : , so the last isomorphism also follows from (2).
Lemma 3.2 gives an idea about how useful to obtain new group-theoretical invariants of the group algebra kG is to find k-canonical subgroup assignations over the family of finite p-groups.This would be also an interesting problem by itself, as it seems to be the most natural way to study how the normal subgroup structure of the group is reflected inside the group algebra.
Let us compare the situation with the case of group algebra with integral coefficients: Let Γ 1 and Γ 2 be two finite groups such that there is an isomorphism φ : ZΓ 1 → ZΓ 2 preserving the augmentation.We adopt temporarily (only for this paragraph) the notations I(Γ i ) and I(N ; Γ i ) for the augmentation ideal of ZΓ i and the augmentation ideal of ZΓ i relative to a normal subgroup N of Γ i .By the so called Normal Subgroup Correspondence in the integral case (see [Seh78,Theorem III.4.17]), φ induces an isomorphism φ * between the lattices of normal subgroups of Γ 1 and Γ 2 .Furthermore, by [Seh78,Theorem III.4.26] this isomorphism satisfies φ(I(N ; Γ 1 )) = I(φ * (N ); Γ 2 ).
Back to the modular case, no such Normal Subgroup Correspondence exists in general.However, there exists a limited version of this correspondence, restricted to the sublattice of the lattice of normal subgroups of G formed by the k-canonical subgroups of G (more properly, evaluations in G of k-canonical assignations).This obvious correspondence is given by φ * (N G ) = N H , for each k-canonical assignation N * .Thus the real problem is to identify the k-canonical subgroups and determine how large this sublattice can be.In general, one might ask the following Question 3.3.Given a field k and a family F of groups, which are the k-canonical subgroup assignations N * : F → Grp?
Unfortunately, if F is the family of finite p-groups, the list of subgroups assignations which are known to be k-canonical is limited to G and to some less known examples appearing in [Sal93] and [BK07], which we will mention later.Even more unfortunately, a choice of subgroup assignation as natural as the center of the group is known to fail to satisfy this property in general, as shown by Bagiński and Kurdics [BK19, Example 2.1], by virtue of a group G of order 81 and maximal class, for which there exists an automorphism φ : , so the terms of the lower central series in general (aside from G = γ 2 (G)) are neither candidates to be k-canonical subgroups assignations.
However, though limited by the existence of this counterexample, it is still possible that the search of this kind of subgroups could lead to new invariants and MIP-related results, or at least to shed some new light on the existing ones.The remainder of the section is devoted to extend this list, and to that end the following pair of easy general facts will be useful.The first one corresponds to [Seh78, Proposition III.6.1].
Lemma 3.4.Let A be an abelian finite p-group, k be a field of characterisitc p, t be a positive integer, and λ : kA → k t (A) be the homomorphism given by x → x p t .Then ker λ = I(Ω t (A); A).
Moreover, the k-linear hull of the image of λ equals k t (A).
Proof.Observe that the p t -power map A → t (A), x → x p t is a surjective homomorphism of groups whose kernel is Ω t (A), so that there is an isomorphism of groups κ : A/Ω t (A) → t (A) which extends to an algebra isomorphism κ : k (A/Ω t (A)) → k t (A).Denote σ : kA → k t (A) the k-linear extension of the p t -power map A → t (A), and π : kA → k (A/Ω t (A)) the canonical projection.Then σ factors as σ = κ • π.Now let τ : kA → kA be the ring homomorphism extending the identity on A and the map x → x p t on k.Then λ factors as λ = τ • σ = τ • κ • π.Since both τ and κ are injective, ker λ = ker π = I(Ω t (A); A).For the last statement it suffices to observe that the image of λ contains t (A).
The following observation is trivial.Lemma 3.5.Let k be a field of characteristic p and G a finite p-group.Let also N and L be normal subgroups of G with N ⊆ L, and let π N : kG → k(G/N ) be the canonical projection.Then N (I(L/N ; G/N )).We close this section with a lemma that allow us to obtain new k-canonical subgroup assignations from known ones, and a series of examples connecting this with other results.A version of this lemma (comprehending only the first item), appears in [GLdRS22] with the name of Transfer Lemma.We choose to reprove it here, as the method of proof is strongly related to the one of the other items.Lemma 3.6 (Transfer Lemma).Let p be a prime number and G and H be finite p-groups.Let t be a positive integer and for Γ ∈ {G, H} let N Γ and L Γ be normal subgroups of Γ such that Γ ⊆ N Γ .Let k be a field of characteristic p and φ : kG → kH be a k-algebra isomorphism preserving the augmentation such that Proof.Throughout the proof, for a normal subgroup K of Γ, we denote by π K : kΓ → k(Γ/K) the natural projection, with kernel I(K; Γ).The hypothesis φ (I(N G ; G)) = I(N H ; H) yields that the isomorphism φ : k(G/N G ) → k(H/N H ) induced by φ makes the following square commutative be the p t -power map, which is a ring homomorphism because of the commutativity of Γ/N Γ .There is a commutative square where the vertical arrow on the right is just the restriction of φ.
(2) Observe that the hypotheses imply that φ( Thus we can assume without loss of generality that N Γ ⊆ L Γ .Now observe that π NΓ (I(L Γ ; Γ)) = I(L Γ /N Γ ; Γ/N Γ ).Hence, by the commutativity of (3.2), Now the result follows from Lemma 3.5 and the commutativity of (3.2) as in the previous item.
(3) Using the decomposition of the center (3.1), we get that where the second equality is due to Lemma 3. Examples 3.7.We illustrate some uses of the Transfer Lemma and relate it to some (recent and classical) known results, as a unified way to approach them.Let k be a field of characteristic p, G and H finite p-groups, and φ : kG → kH be an isomorphism preserving the augmentation, and fix t ≥ 1.
(1) Taking t = 1 and N G = G , Lemma 3.6.(2)yields that This was already proven in [BK07] in order to show that if G is of nilpotency class 2 and KG ∼ = kH, then H also has nilpotency class 2. (2) Taking N Γ = Γ and L Γ = Γ, Lemma 3.6(2) yields that H /H , the two remaining invariants.Some other invariants resembling the previous ones but not appearing in the mentioned theorem also follow readily.For example, for L Γ = Z(Γ)Γ and N Γ = Γ , Lemma 3.6(2) yields φ(I( t (Z(G))G ; G)) = I( t (Z(H))H ; H).So that applying Lemma 3.2(4) we derive that H /H .These two last invariants, as far as we know, were never considered before.(4) For N Γ = Z(G)G , Lemma 3.6(1) and (3.4) yield that φ This leads, with the help of Lemma 3.2, to a number of invariants of the group algebra, which, to the best of our knowledge, were used for the first time in [GLdRS22].There it is shown (see [GLdRS22,Lemma 4.1]) that for p-groups with cyclic derived subgroup and p > 2 there exists an integer t depending only on kG such that C G (G ) = Ω t (G : Z(G)G ), so over this family of groups the subgroup assignation G → C G (G ) is k-canonical.

Abelian direct factors
This section is devoted to prove our main results, and, after a short introduction, it is divided in two parts: the first one is focused on Proposition C, while the second one contains the proof of Theorem A.
Let G be a finite p-group.A homocyclic decomposition d of G is an internal direct product decomposition where U i , A i are subgroups of G, U i is non-abelian and indecomposable, and A i is homocyclic of exponent p i and rank r i , i.e., A i ∼ = (C p i ) ri .Here we allow r i = 0, in which case A i = 1.Such a decomposition always exists by the Krull-Remak-Schmidt theorem.Given a decomposition (4.1), denote does not depend on d, also by the Krull-Remak-Schmidt theorem.Sometimes we are only interested in the isomorphism type of H d i (G), and in such case we drop d from the notation.We say that H i (G) is the homocyclic component of G of exponent p i .Moreover, if we can express G as an internal direct product G = S ⊕ T , where T is homocyclic of exponent p i , then G has a homocyclic decomposition d satisfying T ⊆ H d i (G).In this case we say that d extends the decomposition G = S ⊕ T .With the notation above, we also denote Ab The same considerations using the Krull-Remak-Schmidt theorem yield that the isomorphism types of these subgroups do not depend on d.Hence we can drop it from the notation and write so this notation agrees with the one used in the introduction.
For the rest of the section, let t be a positive integer.The map ). Assume by contradiction that the inclusion is strict.Then there is a non-trivial subgroup ), so that A must be homocyclic with exponent p t .Take an element x ∈ A with order p t .Then there is some element y ∈ Ω t (Z(G)) such that π(y) = x.The order of y is also p t .If y ∈ Φ(G) then The following is the same as Proposition C.
In particular the previous lemma is equivalent to H t (G) ∼ = H t (H) for each t, provided that kG ∼ = kH.
4.2.Proof of Theorem A. We will need another pair of lemmas about homocyclic components, as well as to recover some ideas from [MSS21].
, to show that it is homocyclic of exponent p t it suffices to prove that every element in T \ Φ(T ) = T \ Φ(G) has order greater than p t−1 , and that is a direct consequence of the injectivity of λ t−1 G .We claim that T ∩ t (G)G = 1.Indeed, if 1 = x ∈ T ∩ t (G)G then there exist some y ∈ G and z ∈ G such that x = y p t z.As x ∈ T ⊆ Ω t (Z(G)), there is some integer e and some element g ∈ T \ Φ(T ) = T \ Φ(G) such that g p e = x, with e < t.Therefore g p t−1 = x p t−1−e = (y p t z) p t−1−e = y p 2t−e−1 z for some z ∈ G , that is, g p t−1 ∈ t (G)G , thus 1 = gΦ(G) belongs to the kernel of λ t−1 G , a contradiction.So the claim follows, and we can apply Lemma 4.2 to derive the result.Lemma 4.7.Let T be a subgroup of Ω t (Z(G)) such that d(T ) = d(T Φ(G)/Φ(G)).Then the following conditions are equivalent: Proof.The right-to-left inclusion is trivial, so we prove the converse one.Let π N : kG → k(G/N ) be the natural projection and assume that g ∈ We will use that the following map , is well defined.Moreover there are injective maps From now on we assume that k = F p .Then ψ 1 is an isomorphism (see, for example, [Seh78, Proposition III.1.15]).Let N be a normal subgroup of G. Therefore ψ 1 (N Φ(G)/Φ(G)) = (I(N ; G) + I(G) 2 )/I(G) 2 .Furthermore, from Lemma 4.10 it follows easily that where λt−1 G is given by the p t−1 -power map in the group, xΦ(G) → x p G 0 = G, H 0 = H and φ 0 = φ.Assume that for i ≥ 0 the tuple (G i , H i , φ i ) is defined such that φ i : kG i → kH i is an isomorphism preserving the augmentation.Choose a homocyclic decomposition d i of G i and take a subgroup G i+1 of G i such that G i = G i+1 ⊕ H di i (G i ); then by Lemma 4.12 there is a homocyclic decomposition di of H i such that, given a subgroup H i+1 of H i satisfying H i = H i+1 ⊕ H di i (H i ), one has that kH i = φ(I(H di i (G i ); G i )) ⊕ kH i+1 .Since kG i = I(H di i (G i ); G i ) ⊕ kG i+1 by Lemma 4.8 for n = 1, it follows that φ i induces an isomorphism φ i+1 : kG i+1 → kH i+1 , and we can advance to the next step of the sequence.If i is large enough (for example if p i exceeds the exponent of G), by the Krull-Remak-Schmidt theorem G i ∼ = NAb(G) and H i ∼ = NAb(H).Thus we have an isomorphism φ i : k (NAb(G)) → k (NAb(H)) as desired.
Remarks 4.14.Observe that both Proposition C and Theorem A rely on the results in Section 3: indeed, the former depends on the fact that the subgroup assignations G → Ω t (Z(G)) t (G)G and G → t (G)G are k-canonical to show that the isomorphism type of the quotient Ω t (Z(G)) t (G)G / t (G)G is determined by kG in Proposition 4.5, while the latter, in addition, uses that G → Ω t (Z(G))G is k-canonical to guarantee that the arguments involving the commutative diagram (4.6) make sense.
In the light of Theorem A, Corollary B follows easily.Indeed, it suffices to observe that if a non-abelian finite p-group G satisfies one of the properties (1)-(8) in Corollary B, so does NAb(G), and then apply Proposition 2.1.

Theorem A .
Let k = F p and G and H be finite p-groups.Then kG ∼ = kH if and only if k (NAb(G)) ∼ = k (NAb(H)) and Ab(G) ∼ = Ab(H).

Lemma 3. 1 .
Let k be a field of characteristic p and G a p-group.If N and L are normal subgroups of G, then I(L; G) ∩ I(N ) = I(L ∩ N ; N ).

/
t−1 D p t−1 +1 (G)N .Observe that the image of λt−1 G is contained, by definition, in t−1 (G).Moreover, restricting this diagram to Ω t(Z(G))Φ(G)/Φ((Z(G))G ;G)+I(G) 2 I(G) 2 Λ t−1 G / / I(G) p t−1 +I(N ;G) I(G) p t−1 +1 +I(N ;G) commutes, where ψ 1 is still an isomorphism.Now set N = t (G)G .As t (G) ⊆ t−1 (G) and D p t−1 +1 (G) ⊆ t (G)G , we have that λt−1 G = λ t−1 Gand the previous diagram becomes:/ I(G) p t−1 +I( t(G)G ;G) I(G) p t−1 +1 +I( t(G)G ;G) .(4.3)Observe that by the election of N the diagram can be seen as constructed from the commutative group algebra k(G/G ), so that both λ t−1 G and Λ t−1 G are homomorphisms of elementary abelian groups/vector spaces over k.Lemma 4.11.Let V G be a subspace of I(G) containing I(G) 2 .With the notation above, the following conditions are equivalent:(1) There is a direct sum decompositionV G I(G) 2 ⊕ ker Λ t−1 G = I(Ω t (Z(G))G ; G) + I(G) 2 I(G) 2 .(4.4) 1, a contradiction.Hence the rank of ( y ⊕ H d t (G))Φ(G)/Φ(G) is the rank of y ⊕ H d t (G).Therefore by Lemma 4.2 there exists a subgroup S of G such that G = S ⊕ y ⊕ H d t (G).But now we can extend this decomposition to a homocyclic decomposition d of G verifying that y ⊕ H d