Remarks on Conjectures in Block Theory of Finite Groups †

: In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of ﬁnite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a ﬁnite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of ﬁnite groups, block theory, and character theory are used to achieve these results


Introduction
Let G be a finite group and p be a prime divisor of |G|.Let B be a p-block of group G with defect group D. We consider the triple (k, R, F) to be a p-modular system [1][2][3].This system comprises a complete discrete valuation ring R with a field of fractions k of characteristic 0, where k contains all the primitive |G| th roots of unity.We denote υ p as a valuation on the field k such that υ p (p) = 1.Next, there is the residual field F = R/J(R) of characteristic p, where J(R) is the Jacobson radical of ring R. We can use the field k as a splitting field and F as an algebraically closed field.Let Irr(G) be the set of all ordinary irreducible characters of G, which corresponds to the set of all simple kG-modules.Let M be a simple kG-module, affording the irreducible character ψ of G.Then, there exists an RG-lattice L such that k ⊗ R L = M, but L is not uniquely determined up to isomorphism (see [1,4]).In this case, L is said to be a full RG-lattice in M, and, according to ( [1], Chapter 2, Exercise 16.7), L is an indecomposable RG-lattice.Recall that an RG-lattice L is a left RG-module that has a finite R-basis.Let K(B) be the number of ordinary irreducible characters of B and IBr(G) be the set of all irreducible Brauer characters of G.We use ψ 0 to denote the restriction of the ordinary irreducible character ψ to the set of all p-regular elements (p does not divide the order of the elements) of G. Let L(B) be the number of irreducible Brauer characters of B. We define BL(G) as the set of all p-blocks of G.We use = G to refer to equivalence up to G-conjugacy.
Consider the order of the finite group G to be |G| = p α m such that g.c.d.(p, m) = 1, α, m ∈ Z + for a fixed prime number p. Let ψ ∈ Irr(G).As is well-known, the degree of ψ divides the order of G, as demonstrated in ([5], Theorem 2.4) and ( [6], Theorem 3.11).If In 1955, R. Brauer [10] conjectured that "the defect groups of a p-block B are abelian if and only if all irreducible characters in B have height zero."This conjecture is called Brauer's height zero conjecture (BHZC) and is considered to be one of the most challenging and fundamental conjectures in the representation theory of finite groups, having a significant impact on group theory research.Over the past few decades, several authors have contributed to proving the "only if" implication of BHZC.First, in 1961, P. Fong [11] proved the "only if" implication of BHZC for principle blocks.He also proved the "if" implication of BHZC for the p-solvable group.Later, in [12], he proved the "only if" implication of BHZC for the solvable groups, where the prime number is the largest divisor of the group order.Then, the proof of BHZC was completed for solvable groups in [13,14].In 1984, D. Gluck and T. R. Wolf [15] proved the "only if" implication of BHZC for the p-solvable group.More recently, in 2012, G. Navarro and P. H. Tiep [16] proved the "only if" implication of BHZC for a 2-block B with a Sylow 2-subgroup as a defect group of B. In 2013, R. Kessar and G. Malle [17] proved the "if" implication of BHZC for all finite groups after decades of other contributions on the subject.The next year, B. Sambale [18] investigated BHZC in the case of p-blocks of finite groups with metacyclic defect groups.He proved that BHZC holds for all 2-blocks with defect groups of order 16 at most.Very recently, in 2021, G. Malle and G. Navarro [19] proved the "only if" implication of BHZC for the principle p-block for all prime numbers.After that, the proof of BHZC was completed by proving the "only if" implication of BHZC for any odd prime (see [20]).

Literature Review of Robinson's Conjecture
In 1996, G. Robinson [21] submitted a proposal for the expansion of BHZC, comparing the order of the center of a defect group of a p-block and the p-part of characters' degrees that belong to the p-block of a finite group G: Robinson's conjecture.Suppose G is a finite group.Let χ ∈ Irr(G), which belongs to a p-block B of G with a defect group D.Then, p de f (χ) ≥ |Z(D)|.Moreover, the equality holds if and only if D is abelian.
The other form of RC comes from the relation between the p-defect of the irreducible character χ and the height of χ: The equality in RC holds if and only if D is abelian.If D is abelian, then D = Z(D) according to ([22], Section 2.2, Example (1)), which implies that all irreducible characters in B have height zero from (3).Then, we obtain the "if" implication of BHZC; hence, RC is an expansion for this implication of BHZC.In 1998, M. Murai [23] introduced a reduction of RC to p-blocks of the covering groups for all primes p ≥ 3.In 2014, B. Sambale [18] investigated RC in the case of p-blocks of finite groups with metacyclic defect groups.He proved that RC holds for all 2-blocks with a defect group of order 16 at most.Recently, in 2018, Z. Feng, C. Li, Y. Liu, G. Malle, and J. Zhang [24] proved that RC holds for all primes p ≥ 3 for all finite groups using Murai's reduction of RC.Later, they proved [25] that RC holds using Murai's reduction in the case p = 2 of finite quasi-simple classical groups.Thus, to complete the proof of RC, it only remains to investigate the so-called isolated 2-blocks of the covering groups of exceptional Lie type in the case of an odd characteristic.

Literature Review of Olsson's Conjecture
In [26], J. B. Olsson conjectured that "K 0 (B) ≤ [D B : DB ]", where D B is the defect group of the p-block B of G and DB denotes the commutator subgroup of D, called Olsson's conjecture (OC).The definition of the commutator subgroup can be found in [22,27,28].This conjecture has been proven under certain conditions, but it remains open in general.For instance, in [29], B. Külshammer showed that OC for the p-block B can be derived from the Alperin-Mckay conjecture for B. The same result appeared in [30,31].We remind the reader that the Alperin-Mckay conjecture states that K 0 (b) = K 0 (B), where b is the Brauer correspondent of the p-block B in RN G (D B ).The meaning of the Brauer correspondent of the p-block can be found in [2,5,32,33].However, OC is satisfied for p-solvable, alternating, or symmetric groups in [34][35][36].If D B is the abelian group, then the commutator DB is the trivial subgroup {1 D }.Thus, OC leads to Brauer's K(B) conjecture.Recall that Brauer's K(B) conjecture predicts that K(B) ≤ |D B |; see [37].In particular, OC holds if D B is metacyclic (see [38,39]) or if D B is minimal non-abelian and p = 2 (see [40]).In [41,42], S. Hendren proved OC for some p-block with a defect group that is an extraspecial p-group of order p 3 and exponents p and p 2 .Recently, the authors of [43] proved that OC is fulfilled for controlled blocks with certain defect groups.Furthermore, in the same paper [43], they used the classification of a finite simple group to verify OC for defect groups of p-rank 2 and cases where p > 3 for a minimal non-abelian defect group.
The following example appeared in [37]: Example 1.Let G = S 4 be the symmetric group of degree four.The number of irreducible characters |Irr(S 4 )| = 5.For the case p = 2 We have that the Klein four V 4 is a normal 2-subgroup of S 4 and the centralizer C S 4 (V 4 ) = V 4 .
From ([32], Chapter V, Corollary 3.11), there is only one 2-block B 0 of S 4 with de f (B 0 ) = 3.For the defect group of B 0 , D(B 0 ) ∼ = D 8 , the dihedral group of order 8 is a non-abelian 2-group.Note that there exists χ 3 ∈ Irr(B 0 ) with non-zero height.The center Z(D 8 ) ∼ = C 2 , which is the cyclic group of order 2. We have For the case p = 3 We have the principal 3-block B 0 = {χ 1 , χ 2 , χ 3 }, with de f (B 0 ) = 1.For the defect group of B 0 , D(B 0 ) ∼ = C 3 , which is the cyclic group of order 3.Note that all χ ∈ Irr(B 0 ) are of height zero and satisfy p de f (χ) = |Z(D)| = 3.As D(B 0 ) is an abelian group, the commutator

Anchor Group of Irreducible Characters
Let ψ ∈ Irr(G).Then, ψ may be extended to an algebra map in a unique way with ψ : kG → k.We consider the element which is the unique central primitive idempotent in kG such that ψ(e ψ ) = 0 (see ( [44], Theorem 3.3.1)).As the center Z(RGe ψ ) is a subring of the center Z(kGe ψ ), the algebra RGe ψ is a primitive G-interior R-algebra (see [4]).
The anchor group of an irreducible character appeared for the first time in [45], defined as the defect group of the primitive G-interior R-algebra RGe ψ for any irreducible character ψ of G.As the anchor group of an irreducible character is a defect group, it is a p-subgroup of G (see [46]).
Let us present the most important characteristics of the anchor group of irreducible characters that we use in this paper.The following theorem appears in ([45], Theorems 1.2 and 1.3).
Theorem 1.Consider B to be a p-block of a finite group G with a defect group D. Let ψ ∈ Irr(B) with anchor group A ψ .Suppose L is an RG-lattice affording ψ.The following holds: 1.
The anchor group of ψ is a subgroup of the defect group D (up to G-conjugacy) of B. 2.
The anchor group of ψ contains a vertex of L.

3.
If the defect group D is abelian, then D is an anchor group of ψ.

4.
If ψ has a full defect (height zero), then A ψ is the defect group of B.

5.
If ψ 0 ∈ IBr(G), then L is unique up to isomorphism and A ψ is a vertex of L.

Theorem 2 ([47]
).Let G be a finite group and B be a p-block of G with a defect group D B .Suppose ψ ∈ Irr(B) such that ψ 0 ∈ IBr(B).Then, the anchor group A ψ of ψ is cyclic if and only if the defect group D B is cyclic.In particular, if A ψ is cyclic, then it is the defect group of B.

Lemma 1 ([46]
).Let G be a finite group.If ψ ∈ Irr(G) with a degree prime to p, then the anchor group of ψ is a Sylow p-subgroup of G.

Methodology
Our main methods are based on classical and standard theories on the direct product of finite groups [22,27], block theory [5,32,48], and character theory [6,49].In addition, the Green correspondence theorem is key for studying block theory and calculating the anchor groups of irreducible characters.In fact, given a p-subgroup P of a finite group G, let N G (P) be the normalizer of P in G, Ind(RG|P) be the set of all isomorphism classes of the indecomposable RG-lattices with vertex P, and Ind(RN G (P)|P) be the set of all isomorphism classes of the indecomposable RN G (P)-lattices with vertex P. The following is the Green correspondence theorem, which appears in [1][2][3][4][5]37,50].Theorem 3. Consider the hypotheses in the above paragraph.There is a bijection between Ind(RG|P) and Ind(RN G (P)|P).We say that the lattice L ∈ Ind(RG|P) corresponds to the lattice Ĺ ∈ Ind(RN G (P)|P) if and only if Ĺ is the unique (up to isomorphism) direct summand of the restriction Res G N G (P) (L) with vertex P or L is the unique (up to isomorphism) direct summand of the induction Ind G N G (P) ( Ĺ) with vertex P.
We recall that the vertex of an indecomposable RG-lattice L is a unique (up to Gconjugacy) minimal p-subgroup P of G, such that L is P-projective of G. Consequently, L is a direct summand of the induced Ind G P (N) for some RP-lattice N.

Preliminaries
In this section, we present the classical and standard theories regarding the direct product of finite groups.We detail some characteristics of the ordinary irreducible characters used throughout the paper.
The following propositions are crucial for the representation of direct products of finite groups.
Proposition 1.Let G be a direct product of the finite groups H 1 and H 2 .Let B be a p-block of G with defect group D B .If b i is a p-block of H i with defect group D b i , i = 1, 2, then the following holds: Proof.See ( [48], Propositions 2.3, 2.4, and 2.6).
We offer the classical and standard theories of the direct product of finite groups in the following result.
Proposition 2. Let G be a direct product of the finite groups H 1 and H 2 .Then, the following holds: (a) G is abelian if and only if each of H 1 and H 2 are abelian.
Theorem 4. Let G = H 1 × H 2 be a direct product of the finite groups H 1 and H 2 .Then, Proof.We write ψ ⊗ φ := ψ.φ.See ( [6], Chapter 4, Theorem 4.21).Now, we mention some properties of the ordinary irreducible characters (see ([6], Chapter 2)).The ordinary irreducible character is a homomorphism if it is only linear (i.e., of degree one).Furthermore, the ordinary irreducible character has a kernel.It also has a center, although it is not a group.Definition 1.Consider G to be a finite group and ψ ∈ Irr(G).

•
The center of ψ is a subgroup of G, defined as Z(ψ) Lemma 2. The group G is abelian if and only if every irreducible character of G is of degree one.

Some Conjectures on Direct Products
In this section, we deal with BHCZ, RC, and OC.We prove that the direct product Moreover, the equality holds if and only if D i is abelian for i = 1, 2. We need to show that Per Proposition 2(b), the center of a direct product of groups is the direct product of their centers.Now, from the definition of the defect number of irreducible characters and Theorem 4, we have Therefore, As Proposition 2(a) states, the direct product of finite groups is abelian if and only if each of them is abelian; thus, the equality holds.The other direction is easily achieved through the same steps and citations.
Remark 1.Let B be a p-block of the finite group H 1 × H 2 with a defect group D.Then, from Proposition 1(a), (c), there exists a p-block b i of H i with a defect group We have Now, from Equations ( 4) and ( 5), the height of the irreducible character χ 1 ⊗ χ 2 can be calculated as follows: . Suppose χ has height zero.From Equation ( 6) and the fact that the height of an irreducible character is a non-negative integer by definition, the irreducible characters ψ and φ have height zero.From Proposition 1(b), we can infer that Thus, H 1 and H 2 satisfy OC.The other direction is proven similarly.

Relative Versions of Conjectures and the Green Correspondence Theorem
In this section, we give the conjectures MARC, MHZC, and MAOC, which are related to the algebraic concept of "the anchor group of an irreducible character," which are the relative versions of RC, BHZC, and OC, respectively.By restricting these conjectures to the anchor group instead of the defect group, we prove MARC in some cases.We introduce the relative version of the Green correspondence theorem for a finite simple group G that contains the irreducible character of G with degree p, where p is an odd prime.We give suitable examples of this type of theory.
First, we give the relative version of RC.MARC: Suppose G is a finite group.Let χ ∈ Irr(G) with anchor group A χ .Then, p de f (χ) ≥ |Z(A χ )|, and equality holds if and only if A χ is abelian.
In the following results, we verify MARC in special cases.
If χ has defect n, n ≥ 1.Thus, ; we say that the character ψ is the lift of ψ to G if it satisfies ψ(g) = ψ(gQ), where g ∈ G. From ([49], Theorem 17.3), ψ ∈ Irr(G) if and only if ψ ∈ Irr(G) and ker(ψ) contains Q.So, we have Irr(G) ⊆ Irr(G).From ( [33], p. 137), there exists a unique p-block B of G that contains the p-block B of G, and we write Irr(B) ⊇ Irr(B).Proposition 7. Using the same hypotheses as above, let Q be a normal ṕ-subgroup of G and ψ ∈ Irr(G).Suppose ψ ∈ Irr(G) is the lift of ψ to G. Let ψ 0 ∈ IBr(G) with a cyclic anchor group.If ψ satisfies MARC, then so does ψ.
Proof.Suppose B is a p-block of G that contains ψ and B is a p-block of G that contains ψ.From the details above, Irr(B) ⊆ Irr(B).From ( [33], Theorem 9.9(c)), the defect groups of B and B are isomorphic.Since the anchor group of ψ is cyclic, it is the defect group of B per Theorem 2. Hence, the anchor groups of ψ and ψ are isomorphic.as p de f (ψ) = |G| p ψ(1) p = 1 and de f (ψ) = 0. Hence, per ([3], Theorem 2.3.2),ψ belongs to the singleton p-block, and the defect group of the singleton p-block is the trivial group {1 G }.Then, the result is obtained from Theorem 1 (1).Remark 3. In Theorem 6, we exclude p = 2, as no simple group exists with an irreducible character of degree 2, as in ( [49], Corollary 22.13).
The following corollary immediately follows from Theorem 6.

Corollary 1.
Let G be a simple finite group that has an irreducible character of degree p, where p is an odd prime.If χ ∈ Irr(G) with χ(1) p = p, then the anchor group of χ is the trivial group.
We introduce the relative version of the Green correspondence theorem (Theorem 3) in a simple finite group G, which contains the irreducible character ψ of degree p, where p is an odd prime.Let B be a p-block of G.We define Ind(B|A) to be the set of all isomorphism classes of the indecomposable RG-lattices with vertex A, which belong to B. We write Irr(B|A) := {χ ∈ Irr(B)|χ 0 ∈ IBr(B) and A is the anchor group o f χ}.Lemma 6. Per the same hypotheses as above, let χ ∈ Irr(G) with the non-trivial anchor group A and χ 0 ∈ IBr(G).We write N = N G (A) to be the normalizer of A in G. Let θ ∈ Irr(N) with θ 0 ∈ IBr(N) such that θ lies under ψ; that is, Res G N (ψ), θ = 0.Then, the irreducible characters χ and θ have the same anchor group.However, if χ belongs to the p-block B of G and θ belongs to the p-block b of N, then |Irr(B|A)| = |Irr(b|A)|.
Proof.Assume that L is the indecomposable RG-lattice affording χ and Ĺ is the indecomposable RN-lattice affording θ.Then, from Theorem 1 (5), L is unique up to isomorphism and A is a vertex of L. Per Theorem 6, G possesses a cyclic Sylow p-subgroup that contains all p-subgroups of G. Hence, the vertex of L is equal to the anchor group of an irreducible character χ, which is equal to the defect group of the p-block B (see ( [47], proof of Theorem 5)).Hence, a one-to-one correspondence exists between Irr(B|A) and Ind(B|A).Likewise, there is a one-to-one correspondence between Irr(b|A) and Ind(b|A).The condition Res G N (ψ), θ = 0, is equivalent to Ĺ being a direct summand of the restriction Res G N (L) with vertex A. Per the Green correspondence theorem [1], Ĺ has a vertex A. Thus, Ĺ ∈ Ind(b|A).Therefore, the irreducible character θ has anchor group A, and |Irr(B|A)| = |Irr(b|A)|.
We extracted the Brauer character tables for the following examples from ( [2], Appendix B).These tables can also be obtained for some examples (but not all) from GAP [51].One can also extract the degree of the irreducible characters, the structure of the defect group of a p-block of G, and its normalizer in the group G from GAP [51].
Example 3. Consider G to be a simple group GL(3, 2), the general linear group of order 168 = In the case of p = 3 We have four 3-blocks of G.The principal 3-block B 0 of GL(3, 2) has defect 1 and contains three irreducible characters, all of degree prime to 3. Hence, the anchor group A χ of each irreducible character χ in B 0 is a Sylow 3-subgroup of GL(3, 2) per Lemma 1.The Sylow 3-subgroup of GL(3, 2) is isomorphic to C 3 , a cyclic group of order 3.The two irreducible characters of GL(3, 2) are of degree three, and their anchor groups are the trivial group {1 GL(3,2) } per Theorem 6.The irreducible character ψ of GL(3, 2) with ψ p (1) = 3 has the trivial anchor group per Corollary 1.The normalizer of A χ in GL(3, 2) is N GL(3,2) (A χ ) = S 3 , the symmetric group of degree three.We have that C 3 is a normal 3-subgroup of S 3 and the centralizer C S 3 (C 3 ) = C 3 .From ( [32], Chapter V, Corollary 3.11), there is only one 3-block b 0 of S 3 with de f (b 0 ) = 1 that contains the irreducible character θ lying under χ.Note that |Irr(B 0 |A χ )| = 2 = |Irr(b 0 |A χ )|.The application of the relative versions of the conjectures is detailed in the following: the center of A χ is isomorphic to C 3 , a cyclic group of order 3. Thus, for each χ ∈ Irr(GL(3, 2)), MARC holds because of Proposition 6.As all irreducible characters in the principal 3-block B 0 have height zero, the defect group of B 0 is abelian because of BHZC.Hence, their anchor groups are abelian based on Theorem 1(3).Thus, MHZC holds.As A χ ∼ = C 3 is an abelian group, the commutator Áχ = {1 C 3 }.We have K 0 (B 0 ) = 3 = [C 3 : {1 C 3 }], so MAOC holds.In the case of p = 7 We have two 7-blocks of G.The principal 7-block B 0 of GL(3, 2) has defect 1 and contains five irreducible characters, all of degree prime to 7. Hence, the anchor group A χ of each irreducible character χ in B 0 is a Sylow 7-subgroup of GL (3,2), which is isomorphic to C 7 , a cyclic group of order 7.The singleton 7-block with the trivial defect group , the non-abelian group of order 21.Let b 0 be the principal 7-block of (C 7 : The application of the relative versions of the conjectures is detailed in the following: for each χ ∈ Irr(B 0 ), the center of A χ is isomorphic to C 7 .Then, per Proposition 6, MARC holds.We have that all irreducible characters in the principal 7-block B 0 have height zero.Hence, their anchor groups are abelian, and MHZC holds.As A χ ∼ = C 7 is an abelian group, the commutator Áχ = {1 C 7 }.We have K 0 (B 0 ) = 5 < [C 7 : {1 C 7 }] = 7 , so MAOC holds.
Example 4. Consider G to be a simple group A 5 , the alternating group of degree five of order 60 = 2 2 • 3 • 5.The number of irreducible characters is |Irr(A 5 )| = 5.In the case of p = 3 We have three 3-blocks of A 5 .The principal 3-block B 0 of A 5 has defect 1 and contains three irreducible characters, all of the degree prime to 3. Hence, the anchor group A χ of each irreducible character χ in B 0 is a Sylow 3-subgroup of A 5 , which is isomorphic to C 3 , a cyclic group of order 3.As the two irreducible characters of A 5 are of degree three, their anchor groups are the trivial group {1 A 5 } per Theorem 6.The normalizer of A χ in A 5 is N A 5 (A χ ) = S 3 , the symmetric group of degree three.As in the previous example, there is only one 3-block b 0 of S 3 , which contains the irreducible character θ lying under χ.We have |Irr(B 0 |A χ )| = 2 = |Irr(b 0 |A χ )|.The application of the relative versions of the conjectures is as follows: the center of A χ is isomorphic to C 3 , a cyclic group of order 3. Thus, for each χ ∈ Irr(A 5 ), MARC holds.Note that all irreducible characters in the principal 3-block B 0 have height zero.Hence, their anchor groups are abelian, and MHZC holds.As A χ ∼ = C 3 is an abelian group, the commutator Áχ = {1 C 3 }.We have K 0 (B 0 ) = 3 = [C 3 : {1 C 3 }], and MAOC holds.In the case of p = 5 We have two 5-blocks of A 5 .The principal 5-block B 0 of A 5 has defect 1 and contains four irreducible characters, all of degree prime to 5. Hence, the anchor group A χ of each irreducible character χ in B 0 is a Sylow 5-subgroup of A 5 per Lemma 1.The Sylow 5-subgroup of A 5 is isomorphic to C 5 , a cyclic group of order 5.The normalizer of A χ in A 5 is N A 5 (A χ ) ∼ = D 10 , the dihedral group of order 10.We have that C 5 is a normal 5-subgroup of D Remark 4. If the simple group G does not satisfy the condition stated in Theorem 6, then there is no cyclic Sylow p-subgroup of G, and it does not satisfy Lemma 6, as shown in the following example.
For the following example, we used the Magma computational algebra system [52] to find the Brauer irreducible characters for the group SL (3,3).
Example 5. Let p = 3, G = SL(3, 3) be the special linear group of order 5616.The degrees of the irreducible characters of SL (3,3)  Note that |Irr(SL(3, 3))| = 12, which belong in two 3-blocks.The principal 3-block B 0 has defect 3 and contains 11 irreducible characters, 9 of which are of degree prime to 3 and two of which are of degree 12 and 39, namely, ψ 2 and ψ 12 , respectively.The defect group D of B 0 is the extraspecial 3-group (C 3 × C 3 : C 3 ) of order 27, which is a Sylow 3-subgroup P of G. Thus, from Lemma 1, the anchor group of each irreducible character with degree prime to 3 is a Sylow 3-subgroup.It remains to calculate the anchor groups of ψ 2 and ψ 12 .We have that N The notation in the first row above is as provided in the Atlas of Finite Groups [53].Let L be the indecomposable RG-lattice affording ψ 2 .Let M 1 , M 6 , M 8 , and M 11 be the RN-lattices that afford φ 1 , φ 6 , φ 8 , and φ 11 , respectively.Hence, Res G We can see that M 1 is the direct summand of Res G N (L).Then, per the Green correspondence Theorem 3, the two lattices M 1 and L have the same vertex.We have that the reduction M 1 is the trivial FG-module.Then, per ([54], Corollary 1), M 1 has a Sylow 3-subgroup of N as a vertex.Thus, the Sylow 3-subgroup of N is a vertex of the indecomposable RN-lattice M 1 per ( [2], Chapter 11, Exercise 21).It follows that the Sylow 3-subgroup of N is a vertex of L. We know that the Sylow 3-subgroup of N is equal to the Sylow 3-subgroup P of G in this example.Per Theorem 1(2), the vertex of L is contained in an anchor group of ψ 2 .Therefore, the anchor group of ψ 2 is a Sylow 3-subgroup P of G. To calculate the anchor group of ψ 12 , we use the fact that ψ 3 ∈ Irr(G) is of degree 13.Suppose Ĺ, Ĺ are the indecomposable RG-lattices that afford ψ 12 , ψ 3 , respectively.Consider θ ∈ Irr(N) to be of degree 1, such that Ind G N (θ) = ψ 12 + ψ 3 , as follows: Suppose M is the indecomposable RN-lattice that affords θ.Hence, Ind G N (M) = Ĺ⊕, Ĺ and the two lattices M and Ĺ correspond to each other.Per the Green correspondence theorem, they have the same vertex.As the reduction M of M has dimension prime to 3, the vertex of M is a Sylow p-subgroup of N. As shown in the case of ψ 2 , we conclude that the anchor group of ψ 12 is a Sylow psubgroup P of G.There is only one 3-block b 0 of N. Note that 1 = |Irr(B 0 |P)| = |Irr(b 0 |P)| = 4, which does not satisfy Lemma 6.The application of the relative versions of the conjectures is as follows: the center of the extraspecial 3-group is isomorphic to C 3 , a cyclic group of order 3. Thus, for any χ ∈ Irr(SL(3, 3)), MARC holds.Note that the defect group of B 0 is nonabelian group and there exist ψ 2 , ψ 12 ∈ Irr(B 0 ), which are not of height zero.Thus, MHZC holds.The commutator subgroup of the extraspecial 3-group P is isomorphic to C 3 .We have that K 0 (B 0 ) = 9 = [(C 3 × C 3 : C 3 ) : C 3 ], and MAOC holds.

Discussion
In this paper, we consider the application of RC, BHZC, and OC to a direct product of finite groups.We use classical and standard theories for the direct product of finite groups, block theory, and character theory to accomplish these results.In fact, Propositions 2.3 and 2.6 in [48] are crucial in block theory for a direct product of finite groups.We also discuss the restriction of these conjectures to anchor groups of irreducible characters instead of defect groups.As the anchor group of an irreducible character ψ of a finite group G is a defect group of the primitive G-interior R-algebra RGe ψ , the previous conclusion is logical.We give suitable examples of this reduction.
The review of these conjectures in Sections 1.1-1.3can be compared to our results in a simple finite group.Our discussion revolves around the anchor groups of the irreducible character with degree p, where p is an odd prime.In [50], J. A. Green proved the Green correspondence theorem.In this work, we introduce the relative version of the Green correspondence theorem in a simple finite group G that contains the irreducible character ψ of degree p, where p is an odd prime.To achieve this result, we use Theorem 1 (5): if ψ ∈ Irr(G) such that ψ 0 ∈ IBr(G) with anchor group A ψ , then there is a unique (up to isomorphism) RG-lattice L affording ψ and A ψ is a vertex of L. The outcomes of this paper are important for the modular representation theory of a direct product of finite groups, including an attempt to develop reductions of the RC, BHZC, and OC to the algebraic concept "anchor group of irreducible characters", as well as a relative version of the Green correspondence theorem.We plan to study more conjectures regarding the modular representation of a direct product of finite groups, including an assessment of how reductions can be formed for these conjectures in an attempt to solve them.

Conclusions
This work focuses on BHZC [10], RC [21], and OC [26] (see Sections 1.1-1.3).We prove that the direct product H 1 × H 2 of two finite groups H 1 and H 2 satisfies these conjectures if and only if H 1 and H 2 both satisfy these conjectures.We provide relative versions of RC (MARC), BHZC (MHZC), and OC (MAOC) with respect to the algebraic concept of "the anchor group of an irreducible character."We prove the relative version of RC (MARC) in the case of the center of the anchor group of χ, A χ with order |Z(A χ )| = p and for ψ ∈ Irr(G) with some conditions.Consider G to be a simple finite group.We prove that the anchor group of the irreducible character with degree p is the trivial group, where p is an odd prime.Finally, we present suitable examples of these conjectures and theories in simple finite groups.Many questions and conjectures remain in modular representation theory.We will study more conjectures related to the modular representation of a direct product of finite groups and attempt to develop reductions for these conjectures to solve them.

p n = 1 . 1 .
Literature Review of Brauer's Height Zero Conjecture
the finite groups H 1 and H 2 satisfies these conjectures if and only if H 1 and H 2 satisfy these conjectures.Let G be a direct product of the finite groups H 1 and H 2 .Then, G satisfies RC if and only if H 1 and H 2 satisfy RC.Proof.Suppose H i , i = 1, 2, are finite groups that satisfy RC.If χ i ∈ Irr(H i ), which belongs to a p-block b i of H i with a defect group ⊗ b 2 of a finite group H 1 × H 2 is abelian.Then, D = G D 1 × D 2 and, per Proposition 2(a), the direct product of groups is abelian if and only if each of them is abelian.Thus, the defect group D i of a p-block b i of H i is abelian for i = 1, 2. As H 1 and H 2 satisfy BHZC, for all χ i ∈ Irr(b i ), we have h(χ i ) = 0, i = 1, 2. Then, from Equation (6), we obtain the height of all irreducible characters in the p-block b 1 ⊗ b 2 as zero.For the converse implication, suppose all irreducible characters χ 1 ⊗ χ 2 in the p-block b 1 ⊗ b 2 of H 1 × H 2 have height zero.From Equation (6) and the fact that the height of an irreducible character is a non-negative integer by definition, we find that all irreducible characters in a p-block b i of H i for i = 1, 2 have height zero.Hence, per BHZC, the defect group D i of a p-block b i of H i is abelian for i = 1, 2. Now, also per Proposition 2(a), the defect groupD 1 × D 2 of the p-block b 1 ⊗ b 2 of H 1 × H 2 is abelian.The same steps and citations can also be used to obtain the result in the other direction.Let G be a direct product of the finite groups H 1 and H 2 .Then, G satisfies OC if and only if H 1 and H 2 satisfy OC.Proof.Suppose B ∈ BL(G) with defect group D B .From Proposition 1(a), (c), there exists a p-block b i of H i with a defect group D b i Proposition 4. Let G be a direct product of the finite groups H 1 and H 2 .Then, G satisfies BHZC if and only if H 1 and H 2 satisfy BHZC.Proof.Suppose H i , i = 1, 2, are finite groups that satisfy BHZC.Let D i be a defect group of a p-block b i of H i for i = 1, 2. Suppose the defect group D of a p-block b 1 are which is the group of order 108.We can see that Res G N (ψ 2 ) = 2φ 1 + φ 6 + φ 8 + φ 11 , where φ 1 , φ 6 , φ 8 , φ 11 ∈ Irr(N), as follows: