Group algebras in which the socle of the center is an ideal

. Let F be a field of characteristic p > 0. We study the structure of the finite groups G for which the socle of the center of FG is an ideal in FG and classify the finite p -groups G with this property. Moreover, we give an explicit description of the finite groups G for which the Reynolds ideal of FG is an ideal in FG .


Introduction
Let F be a field and consider the group algebra F G of a finite group G and its center ZF G.The question when the Jacobson radical of ZF G is an ideal in F G has been answered by Clarke [4], Koshitani [7] and Külshammer [9].We now study the corresponding problem for the socle soc(ZF G) of ZF G as well as for the Reynolds ideal R(F G) of F G. In a prequel to this paper [3], we have already given some approaches to these problems for general symmetric algebras.Now, our aim is to analyze the structure of the finite groups G for which soc(ZF G) or R(F G) are ideals of F G in a group-theoretic manner.For the Reynolds ideal, we obtain the following characterization: Theorem A. Let F be a field of characteristic p > 0 and let G be a finite group.Then the Reynolds ideal R(F G) is an ideal in F G if and only if G ′ is contained in the p-core O p (G) of G.
As a consequence of this result, it follows that if soc(ZF G) is an ideal in F G, one has G = P ⋊ H for a Sylow p-subgroup P of G and an abelian p ′ -group H. Based on this decomposition, we derive some fundamental results on the structure of finite groups G for which soc(ZF G) is an ideal in F G. Subsequently, we classify the finite p-groups G with this property: Theorem B. Let F be a field of characteristic p > 0 and let G be a finite p-group.Then soc(ZF G) is an ideal in F G if and only if This statement will allow us to restrict our investigation to the case P = G ′ .A detailed analysis of the structure of finite groups G for which soc(ZF G) is an ideal in F G, based on the above results, will be carried out in a sequel to this paper.
We proceed as follows: First, we introduce our notation (see Section 2) and study the general structure of the finite groups G for which soc(ZF G) or R(F G) are ideals in F G (see Section 3).In Section 4, we classify the p-groups G for which soc(ZF G) is an ideal in F G for a field F of characteristic p > 0. In Section 5, we derive the decomposition of G given in Theorem D.

Notation
Let G be a finite group and p a prime number.As customary, let G ′ , Z(G) and Φ(G) denote the derived subgroup, the center and the Frattini subgroup of G, respectively.For elements a, b ∈ G, we define their commutator as [a, b] = aba −1 b −1 .We write [g] for the conjugacy class of g ∈ G and set Cl(G) to be the set of conjugacy classes of G.The nilpotency class of a nilpotent group G will be denoted by c(G).Recall that every p-group is nilpotent.For subsets S and T of G, let C T (S) and N T (S) denote the centralizer and the normalizer of S in T , respectively.As customary, let O p (G), O p ′ (G) and O p ′ ,p (G) be the p-core, the p ′ -core and the p ′ , p-core of G, respectively.By O p (G) and O p ′ (G), we denote the p-residual subgroup and the p ′ -residual subgroup of G, respectively.As customary, let g p and g p ′ be the p-part and the p ′ -part of an element g ∈ G, respectively.The p ′ -section of g is given by all elements in G whose p ′ -part is conjugate to For a field F and a finite-dimensional F -algebra A, we denote by J(A) and soc(A) its Jacobson radical and (left) socle, the sum of all minimal left ideals of A, respectively.Both J(A) and soc(A) are ideals in A. In this paper, an ideal I of A is always meant to be a two-sided ideal, and we denote it by I ⊴ A. Additionally, we study the Reynolds ideal R(A) := soc(A) ∩ Z(A) of A. Furthermore, let K(A) denote the commutator space of A, that is, the F -subspace of A spanned by all elements of the form ab − ba with a, b ∈ A.
In the following, we consider the group algebra F G of G over F .Recall that F G is a symmetric algebra with symmetrizing linear form (2.1) For subsets S and T of F G, we write lAnn T (S) and rAnn T (S) for the left and the right annihilator of S in T , respectively, and Ann T (S) if both subspaces coincide.For H ⊆ G, we set In this paper, we mainly study the Jacobson radical J(ZF G) and the socle soc(ZF G) of the center of F G as well as the Reynolds ideal R(F G).All three spaces are ideals in ZF G, but not necessarily in F G. Note that J(ZF G) = J(F G) ∩ ZF G holds (see [10,Theorem 1.10.8])and that by [10,Theorem 1.10.22],we have soc(ZF G) = Ann ZF G (J(ZF G)).Furthermore, observe that J(ZF G), soc(ZF G) and R(F G) are ideals in F G if and only if they are closed under multiplication with elements of F G since they are additively closed.
We recall the definition of the augmentation ideal If F is a field of characteristic p > 0 and G is a p-group, then J(F G) and ω(F G) coincide (see [10,Theorem 1.11.1]).For a normal subgroup N of G, we consider the canonical projection Its kernel is given by ω [10,Proposition 1.6.4]).

General properties
Let F be a field.In this part, we answer the question for which finite groups G the Reynolds ideal R(F G) is an ideal in F G.Moreover, we derive structural results on finite groups G for which soc(ZF G) is an ideal in F G. In the next section, these will be applied in order to classify the finite groups of prime power order with this property.
Concerning the choice of the underlying field F , we note the following: Remark 3.1.
(i) Assume that F is of characteristic zero or of positive characteristic not dividing |G|.By Maschke's theorem, the group algebra F G is semisimple.In particular, From now on until the end of this paper, we therefore assume that F is an algebraically closed field of characteristic p > 0.
This section is organized as follows: We first derive a criterion for soc(ZF G) ⊴ F G (see Section 3.1) and answer the question when the Reynolds ideal of F G is an ideal in F G (see Section 3.2).In Section 3.3, we investigate p-blocks of F G. Subsequently, we find a basis for J(ZF G) (see Section 3.4) and construct elements in soc(ZF G) arising from normal p-subgroups of G (see Section 3.5).In Section 3.6, we study the case that G ′ is contained in the center of a Sylow p-subgroup of G.We conclude this part by investigating the transition to quotient groups in Section 3.7 and studying central products in Section 3.8.

Criterion for soc(ZF G) ⊴ F G.
Let G be a finite group.In this section, we derive an equivalent criterion for soc(ZF G) ⊴ F G.  [11,Lemma 3.1.2]).□

Reynolds ideal.
Let G be a finite group.In this section, we answer the question when the Reynolds ideal R(F G) is an ideal in F G. Our main result is the following: Theorem 3.4.The following properties are equivalent: (iii) G = P ⋊ H with P ∈ Syl p (G) and an abelian p ′ -group H.
In this case, we have Thus, for g ∈ G ′ , the element g − 1 is nilpotent.Hence there exists n ∈ N with 0 = (g − 1) p n = g p n − 1.This shows that G ′ is a p-group and hence contained in O p (G).Now assume G ′ ⊆ O p (G) and let P ∈ Syl p (G).Then G ′ ⊆ P follows, so P is a normal subgroup of G and G/P is abelian.By the Schur-Zassenhaus theorem, P has a complement H in G.Moreover, H is isomorphic to G/P and thus abelian.
Finally suppose that G = P ⋊ H holds, where P ∈ Syl p (G) and H is an abelian p ′ -group.In particular, we have This proves Theorem A. Moreover, we obtain the following necessary condition for soc(ZF G) ⊴ F G:  [5,Theorem 5.3.6] and that this is a normal subgroup of × P , and we conclude that C G (P ) = O p ′ (G) × Z(P ) holds.(iv) Since R(F G) is spanned by the p ′ -section sums of G (see [8,Equation (39)]), every p ′ -section is of the form hP for some h ∈ H.

3.3.
Blocks and the p ′ -core.Let G be an arbitrary finite group.In this section, we investigate the conditions soc(Z(B)) holds for all i ∈ {1, . . ., n}, and the analogous statement is true for the Reynolds ideal.Furthermore, it is known that the principal blocks of F G and F Ḡ are isomorphic for Ḡ := G/O p ′ (G).
For the Reynolds ideal, we obtain the following result: Lemma 3.8.The following are equivalent: (i) There exists a block Proof.Assume that (i) holds.By [9, Proposition 4.1], this implies B ∼ = B 0 and hence (ii) holds.Now assume that (ii) holds.By [9, Remarks 2.2 and 3.1], every simple B 0 -module is one-dimensional.Since the intersection of the kernels of the simple B 0 -modules is given by Proof.As in the proof of Lemma 3.8, the equivalence of (i) and (ii) follows by [9, Proposition 4.1] and the equivalence of (ii) and (iii) follows from the fact that B 0 and B0 are isomorphic.□ This has the following important consequence:

Basis for J(ZF G).
Let G = P ⋊ H be a finite group with P ∈ Syl p (G) and an abelian p ′ -group H (see Theorem 3.4).The aim of this section is to determine an F -basis for J(ZF G).In the given situation, the kernel of the canonical map ν P : F G → F [G/P ] is given by J(F G) (see [10,Corollary 1.11.11]).In the following, we distinguish two types of conjugacy classes: Remark 3.12.Let C ∈ Cl(G).We obtain | C| = 1 for the image C ∈ Cl(G/P ) of C in G/P since this group is abelian.Now two cases can occur: • |C| is divisible by p: Note that the basis of J(ZF G) given in Theorem 3.14 consists of homogeneous elements with respect to this grading.In particular,

Elements in soc(ZF G).
Let G be an arbitrary finite group.In this section, we study elements of soc(ZF G) which arise from certain normal p-subgroups of G. Using these, we show that G ′ has nilpotency class at most two if soc(ZF G) is an ideal in F G.Moreover, we derive a decomposition of G which will later be used to prove Theorem D.
Lemma 3.16.Let N be a normal p-subgroup of G and set Proof.Note that M is a normal subgroup of G. Let R be an orbit of the conjugation action of N on C and consider an element r ∈ R.
In particular, we have The following result will be particularly useful for our derivation on p-groups: Proof.By Lemma 3.18, we obtain (Z(P )M ) + ∈ soc(ZF G) for M = {x ∈ [P, G] : In particular, this implies (Z(P )G ′ ) + ∈ soc(ZF G).Since we have     G) is an ideal in F G. We conclude this section with a result on p-groups, which is an immediate consequence of Lemma 3.18: In particular, G ′ is abelian.□ 3.6.Special case G ′ ⊆ Z(P ).Let G = P ⋊ H be a finite group with P ∈ Syl p (G) and an abelian p ′ -group H.In this section, we show that soc(ZF G) is an ideal in F G if G ′ ⊆ Z(P ) holds.
(i) By Remark 3.6, gP is a p ′ -section of G.In particular, [h] is the unique p ′ -conjugacy class contained in gP and hence [g p ′ ] = [h] follows.Since H is abelian, we have g p ′ = uhu −1 for some u ∈ P .Due to g p ∈ Z(P ), this yields g = uhg p u −1 and hence [g] = [hg p ].We may therefore assume g p ′ = h.For x = p x h x with p x ∈ P and h x ∈ H, we have

6). Since h and h ′ commute, we obtain hu
Proof.We may assume u ̸ = 1.By Remark 3.6, m : since the elements in h[u] are conjugate by Lemma 3.22 (ii).Since p does not divide m, we obtain Proof.Consider an element y = g ∈ G a g g ∈ soc(ZF G).Let g ∈ G and write g = cz with c ∈ C G (H) and z ∈ Z(P ).By Lemma 3.24, we have a g = a cz = a c .Hence y ∈ Z(P This proves the first part of Theorem C. The next example shows that the condition G ′ ⊆ Z(P ) is not necessary for soc(ZF G) ⊴ F G.
Example 3.26.Let F be an algebraically closed field of characteristic p = 3 and consider the group G = SmallGroup(216, 86) in GAP [12].We have G = G ′ ⋊ H, where G ′ is the extraspecial group of order 27 and exponent three, and H ∼ = C 8 permutes the nontrivial elements of G ′ /G ′′ transitively and acts on G ′′ = Z(G ′ ) by inversion.In particular, G ′ is nonabelian.For h ∈ H, we set S h := soc(ZF G) ∩ F hG ′ .Due to the H-grading of F G introduced in Remark 3.15, it suffices to show S h = F (hG ′ ) + for all h ∈ H. Clearly, we have (hG ′ ) + ∈ S h .The derived subgroup G ′ decomposes into the G-conjugacy classes {1}, G ′′ \{1} and G ′ \G ′′ .For 1 ̸ = h ∈ H, the coset hG ′ consists of a single conjugacy class for ord(h) = 8 and of two conjugacy classes for ord(h) ∈ {2, 4}.In the first case, we directly obtain S h = F (hG ′ ) + .In the latter case, we have

Quotient groups.
Let G be a finite group of the form P ⋊ H with P ∈ Syl p (G) and an abelian p ′ -group H.We fix a normal subgroup N ⊴ G with quotient group Ḡ := G/N .Our aim is to study the transition to the group algebra F Ḡ. The image of an element g ∈ G in Ḡ will be denoted by ḡ (similarly for subsets of G).Note that Ḡ is of the form P ⋊ H with P ∈ Syl p ( Ḡ) and the abelian p ′ -group H.In the following, we consider the canonical projection map together with its adjoint map ν * N : F Ḡ → F G, which is defined by requiring λ(ν * N (x)y) = λ(xν N (y)) for all x ∈ F Ḡ and y ∈ F G. Here, λ and λ denote the symmetrizing linear forms of F G and F Ḡ given in (2.1), respectively.It is easily verified that ν * N is given by ν

Note that ν *
N is a linear map with image N + •F G and that it is injective as ν N is surjective.Remark 3.27.For a ∈ F Ḡ, it is easily seen that a ∈ ( Ḡ′ ) Now let N again be an arbitrary normal subgroup of G.We obtain the following necessary condition for soc(ZF G) ⊴ F G: Theorem 3.31.We have Central products will play an important role throughout our investigation, for instance in the decomposition of G given in Theorem D. We conclude this part with a generalization of Theorem 4.14 to arbitrary finite groups, which is a stronger variant of Theorem 3.25:

Decomposition of G into a central product
Let F be an algebraically closed field of characteristic p > 0. We consider an arbitrary finite group G for which soc(ZF G) is an ideal in F G. By Theorem 3.4, we may write G = P ⋊ H with P ∈ Syl p (G) and an abelian p ′ -group H.In this section, we prove Theorem D. Combined with the results on p-groups from the last section, it reduces our investigation to the case that G ′ is a Sylow p-subgroup of G.

Lemma 3 . 9 .
Finally, assume that (iii) holds.Then we have Ḡ′ ⊆ O p ( Ḡ). Theorem 3.4 yields R(F Ḡ) ⊴ F Ḡ, which implies R( B0 ) ⊴ B0 by Remark 3.7.Since B 0 and B0 are isomorphic, we obtain R(B 0 ) ⊴ B 0 .□ Concerning the analogous problem for the socle of the center, we first observe the following: The following are equivalent: (i) There exists a block B of F G for which soc(Z(B)) ⊴ B holds.(ii) For the principal block B 0 of F G, we have soc(Z(B 0 )) ⊴ B 0 .(iii) For the principal block B0 of F Ḡ, we have soc(Z( B0 )) ⊴ B0 .

Remark 3 . 11 .
3] and soc(ZF Ḡ) is an ideal of F Ḡ by[3, Proposition 2.10].For the latter, note thatF Ḡ ∼ = F G/ Ker(ν O p ′ (G)) can be viewed as a quotient algebra of F G. Now let R(F G) and soc(ZF Ḡ) be ideals in F G and F Ḡ, respectively.By Remark 3.7, this yields soc(Z( B0 )) ⊴ B0 and hence soc(Z(B 0 )) ⊴ B 0 (see Lemma 3.9).Since R(F G) is an ideal in F G, all blocks of F G are isomorphic to B 0 by [9, Proposition 4.1].By Remark 3.7, we then obtain soc(ZF G) ⊴ F G.□ Assume that G is of the form G = P ⋊H with P ∈ Syl p (G) and an abelian p ′ -group H. Then soc(ZF G) ⊴ F G is equivalent to soc(ZF Ḡ) ⊴ F Ḡ (see Theorem 3.4 and Lemma 3.10).By going over to the quotient group G/O p ′ (G), we may therefore restrict our investigation to groups G with O p ′ (G) = 1.
by p: In this case, |P | divides |C G (g)| for any g ∈ C.This yields P ⊆ C G (g) and hence C ⊆ C G (P ).As customary, we decompose g = g p ′ g p into its p ′ -part and p-part.Note that g p ′ ∈ O p ′ (G) ⊆ Z(G) and g p ∈ Z(P ) hold by Remark 3.6.Due tog p ′ ∈ Z(G), we have C = g p ′ [g p ] and the element C + − |C| • g p ′ is contained in Ker(ν P ) ∩ ZF G = J(ZF G).Definition 3.13.For C ∈ Cl(G) with C ̸ ⊆ O p ′ (G), we set b C := C + if p divides |C|, and b C := C + − |C| • g p ′ otherwise.With this, we obtain the following basis for J(ZF G): Theorem 3.14.An F -basis for J(ZF G) is given by B := b C : C ∈ Cl(G), C ̸ ⊆ O p ′ (G) .Proof.By Remark 3.12, we have B ⊆ J(ZF G).Note that the elements in B ∪ O p ′ (G) form an F -basis for ZF G. Since the algebra F O p ′ (G) is semisimple, J(ZF G) is spanned by B. □ Remark 3.15.The decomposition F G = h ∈ H F hP gives rise to an H-grading of F G.
r), which yields |R| = |N : C N (r)| ̸ = 1.Set X := ⟨N, R⟩ = ⟨N, r⟩.First consider the case [N, G] ⊆ Z(N ).Then the map f : N → N, n → [n, r] is a group endomorphism with kernel C N (r).We set S := Im(f ).Then we have |R| = |N : C N (r)| = |S|, so in particular, |S| is a nontrivial power of p.Let Ḡ := G/M and set ḡ := gM ∈ Ḡ for g ∈ G (similarly for subsets of G).Note that R is an orbit of the conjugation action of N on C. As before, we obtain | R| = | N : C N (r)| = | S| = |S : S ∩ M |.Since S ⊆ [N, G] is a nontrivial p-group, |S ∩ M | is divisible by p.With this, we obtain ν M (R + ) = |R| | R| • R+ = |S ∩ M | • R+ = 0. Now we consider the general case.Let L := [N, [N, G]].We set G := G/L and write g := gL ∈ G for g ∈ G (similarly for subsets of G).Note that we have [ N , [ N , G]] = 1 and hence which translates to a g = a gz for all g ∈ G. Hence x ∈ O p (Z(G)) + • F G follows.□ Observe that the right inclusion in the preceding lemma holds for arbitrary finite groups.The next result is the central ingredient in the proof of Theorem D: Proposition 3.20.Suppose that G ′ ⊆ C P (N )N holds for every normal p-subgroup N of G. Then the following hold: (i) We have [P, G ′ ] ⊆ Z(G ′ ).In particular, this implies G ′′ ⊆ Z(P ) and that the nilpotency class of G ′ is at most two.Moreover, we obtain Φ(G ′ ) ⊆ Z(G ′ ).(ii) We have P = C P (H) * [P, H] and G = C P (H) * O p (G).
Proof.(i)Let D be a critical subgroup of P (in the sense of [5, Theorem 5.3.11]).Then D is normal in G, and Z(D) contains Φ(D), C P (D) and [P, D].By assumption, we have G ′ ⊆ DC P (D) = D. Hence we have [ let b C denote the associated element of J(ZF G) (see Definition 3.13) and consider the basis B := {b C : C ∈ Cl(G), C ̸ ⊆ O p ′ (G)} of J(ZF G) (see Theorem 3.14).Clearly, ν N (J(ZF G)) is spanned by the images of the elements in B. We now derive a more convenient generating set.Lemma 3.28.Let C ∈ Cl(G) be a conjugacy class with C ̸ ⊆ O p ′ (G).We have b C / ∈ Ker(ν N ) if and only if C ̸ ⊆ O p ′ ( Ḡ) holds and k := |C|/| C| is not divisible by p.In this case, the basis element bC of J(ZF Ḡ) corresponding to C ∈ Cl( Ḡ) is well-defined and we have ν N (b C ) = k • b C .Proof.Observe that C is indeed a conjugacy class of Ḡ and that ν N (C + ) = k • C+ with k := |C|/| C| holds.Suppose first that p divides |C|, so b C = C + holds.Then ν N (b C ) ̸ = 0 is equivalent to k ̸ ≡ 0 (mod p),and in this case we have | C| ≡ 0 (mod p).Since O p ′ ( Ḡ) ⊆ Z( Ḡ) holds, this implies C ̸ ⊆ O p ′ ( Ḡ).Moreover, we have b C = C+ and thus ν N (b C ) = k • b C .It remains to consider the case C ⊆ C G (P ).There, we have C ⊆ C Ḡ( P ).If C ̸ ⊆ O p ′ ( Ḡ) holds, then b C is defined, and we have b C = C + − |C| • g p ′ and b C = C+ − | C| • ḡp ′ for g ∈ C.This shows that ν N (b C ) = k • b C holds.If, in addition, k ̸ ≡ 0 (mod p), then ν N (b C ) ̸ = 0 follows.Suppose conversely that ν N (b C ) ̸ = 0 holds.We write C = g p ′ D for g p ′ ∈ O p ′ (G) and D ∈ Cl(G) with D ⊆ Z(P ) (see Remark 3.12).Assume that C ⊆ O p ′ ( Ḡ) holds.Then we have D = ḡ−1 p ′ C ⊆ O p ′ ( Ḡ) due to ḡp ′ ∈ O p ′ ( Ḡ).As D consists of p-elements, we must have D = {1}, which yields the contradiction ν N (bC ) = ν N (g p ′ D + − |D| • g p ′ ) = 0.This shows that C ̸ ⊆ O p ′ ( Ḡ) holds.Hence we have ν N (b C ) = k • b C , so that k ̸ ≡ 0 (mod p).□ Definition 3.29.Set Cl p ′ ,N (G) := {C ∈ Cl(G) : C ̸ ⊆ O p ′ (G)and b C / ∈ Ker(ν N )} and let Cl + p ′ ,N (G) := b C : C ∈ Cl p ′ ,N (G) be the set of corresponding basis elements of J(ZF G) (see Definition 3.13).By Cl p ′ ,N (G) ⊆ Cl( Ḡ), we denote the set of images of the conjugacy classes in Cl p ′ ,N (G) and set Cl + p ′ ,N (G) := b C : C ∈ Cl p ′ ,N (G) , where b C denotes the basis element of J(ZF Ḡ) corresponding to C. If N is a p-group, the p ′ -conjugacy classes of length divisible by p in Cl p ′ ,N (G) can be easily characterized: Lemma 3.30.Consider a normal p-subgroup N of G and let C ̸ ⊆ C G (P ) be a p ′ -conjugacy class.Then we have C ∈ Cl p ′ ,N (G) if and only if

Theorem 5 . 1 (
Theorem D).We have G = C P (H) * O p (G).Moreover, soc(ZF C P (H)) and soc(ZF O p (G)) are ideals in F C P (H) and F O p (G), respectively.The socle of ZF G is explicitly given by soc(ZF G) = (Z(P )G ′ ) + • F G.
and only if ZF G = F G holds, that is, if and only if G is abelian.(ii)Let F be a field of characteristic p > 0 and let G be a finite group.Then soc(ZF p G) is an ideal in F p G if and only if soc(ZF G) is an ideal in F G. A similar statement holds for the Reynolds ideal.
this, the claim follows from Theorem 3.4.
Hence B = 1 follows, which yields P = C P (H) * [P, H].By Remark 3.6, this implies G = C P (H) * H[P, H] = C P (H) * O p (G). □ By Lemma 3.18, the properties given in Proposition 3.20 hold whenever soc(ZF Let g ∈ G with g p ∈ Z(P ).For y ∈ ZF G with y • [g p ′ ] + = 0, we have y • [g] + = 0.Proof.The group P acts on [g] by conjugation with orbits of the form [g p ′ ]u with u ∈ P (seeLemma 3.22).In particular, [g] is a disjoint union of sets of this form.Hence y • [g Since the elements of P centralize U h ⊆ [P, G] ⊆ Z(P ) and conjugation with elements of H permutes the elements [a, h] with a ∈ P , it follows that U h is normal in G. □ Sofia Brenner & Burkhard Külshammer Corollary 3.23.