A transfer is a group homomorphism from a finite group to an abelian quotient
group of a subgroup of the group. In this paper, we explain some of the
properties of transfers by using noncommutative determinants. These properties
enable us to understand transfers more naturally

Yamaguchi, Naoya

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A transfer is a group homomorphism from a group to an abelian quotient group of a subgroup of finite index. In this paper, we give a natural interpretation of the transfers in group theory in terms of noncommutative determinants

Naoya Yamaguchi

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Advances in Group Theory and Applicationsc© 2018 AGTA - www.advgrouptheory.com/journal6 (2018), pp. 101–110 ISSN: 2499-1287DOI: 10.32037/agta-2018-007Proof of Some Properties of Transfer UsingNoncommutative Determinants *Naoya Yamaguchi(Received Mar. 31, 2018; Accepted Jul. 11, 2018 — Communicated by F. de Giovanni)AbstractA transfer is a group homomorphism from a group to an abelian quotient groupof a subgroup of finite index. In this paper, we give a natural interpretation of thetransfers in group theory in terms of noncommutative determinants.Mathematics Subject Classification (2010): 20C05, 11H60, 15A15Keywords: transfer; noncommutative determinant; group algebra1 IntroductionA transfer is defined by Issai Schur (see [7]) as a group homomorphismfrom a group to an abelian quotient group of a subgroup of the group. Infinite group theory, transfers play an important role in transfer theorems.Transfer theorems include, for example, Alperin’s theorem (see [1], Theo-rem 4.2), Burnside’s theorem (see [6], Hauptsatz 4.2.6), and Hall-Wielandt’stheorem (see [5], Theorem 14.4.2).On the other hand, Eduard Study defined the determinant of a quater-nionic matrix (see [3]). The Study determinant uses a regular representationfrom Mat(n, H) to Mat(2n, C), where H is the quaternions. Similarly, wedefine a noncommutative determinant. It is similar to the Dieudonnédeterminant (see [2]).* This work is partially supported by a grant from the Japan Society for the Promo-tion of Science (JSPS KAKENHI Grant Number 15J06842)102 Naoya YamaguchiTôru Umeda suggested that a transfer can be derived as a noncommuta-tive determinant (see [8], Footnote 7). In this paper, we develop his ideasin order to explain the properties of the transfers by using noncommutativedeterminants. As a result, we give a natural interpretation of the transfersin group theory in terms of noncommutative determinants.Let G be a group, H a subgroup of G of finite index, K a normal subgroupof H, and the quotient group H/K of K in H an abelian group. The transferof G into H/K is a group homomorphismVG→H/K : G→ H/K.The definition of the transfer VG→H/K uses the left (or right) coset represen-tatives of H in G. We can show that a transfer has the following properties.(1) A transfer is a group homomorphism from G to H/K (see [4], Theo-rem 3.1).(2) A transfer is invariant under a change of coset representatives (see [4],Proposition 3.1).(3) A transfer by left coset representatives equals a transfer by right cosetrepresentatives (see [4], Section 3.1).Let R be a commutative ring with unity and RG the group algebra of Gover R whose elements are all possible finite sums of the form∑g∈Gxgg, xg ∈ R.The noncommutative determinant uses a left (or right) regular representationfrom RG to Mat(m,RH), where m is the index of H in G. Our main result isthe following.Theorem 1.1 We can regard the transfer VG→H/K as the noncommutative deter-minant Det. That is, we haveDet(g) = sgn(g)VG→H/K(g) (g ∈ G)where the mapsgn : G→ {−1, 1}is a group homomorphism and 1 is the unit element of R. In addition, we can showthat the above properties of the transfer (1), (2), and (3) by the following propertiesof the noncommutative determinant Det.(1’) The determinant is a multiplicative map from RG to R(H/K).(2’) The determinant is invariant under a change of a regular representation.(3’) The determinant of any left regular representation equals the determinant ofany right regular representation.Proof of some properties of transfer 1032 Definition of the transferHere, we define the left and right transfer of G into H/K.LetG = t1H ∪ t2H ∪ . . . ∪ tmH.That is, we take a complete set {t1, t2, . . . , tm} of left coset representativesof H in G. We define g = ti for all g ∈ tiH. The definition of the left transferis the following.Definition 2.1 (Left transfer; see Definition 3.3 of [4]) We define the mapVG→H/K : G→ H/KbyVG→H/K(g) =m∏i=1{(gti)−1gti}K.We call the map VG→H/K the left transfer of G into H/K.Next, we define the right transfer of G into H/K.LetG = Hu1 ∪ Hu2 ∪ . . . ∪ Hum.That is, we take a complete set{u1,u2, . . . ,um}of right coset representatives of H in G. We define g̃ = ui for all g ∈ Hui.The definition of the right transfer is the following.Definition 2.2 (Right transfer; see Definition 3.3 of [4]) We define the mapṼG→H/K : G→ H/KbyṼG→H/K(g) =m∏i=1{uig (ũig)−1}K.We call the map ṼG→H/K the right transfer of G into H/K.The definitions of the left and right transfers use the coset representativesof H in G. But, we can show that the left and right transfers are invariantunder a change of coset representatives. Furthermore, we can show thata transfer is a group homomorphism from G to H/K and a transfer by leftcoset representatives equals a transfer by right coset representatives.104 Naoya Yamaguchi3 Definition of the noncommutative determinantHere, we define the noncommutative determinant.First, we define the left regular representation of RG. We take a completesetT = {t1, t2, . . . , tm}of left coset representatives of H in G. Then, for all α ∈ RG, there exists aunique LT (α) ∈Mat(m,RH) such thatα(t1 t2 . . . tm) = (t1 t2 . . . tm)LT (α),where we regard α(t1 . . . tm) as scalar multiplication (αt1 . . . αtm). The R-al-gebra homomorphismLT : RG 3 α 7→ LT (α) ∈Mat(m,RH)is called the left regular representation with respect to T .LetT ′ = {t ′1, t′2, . . . , t′m}be an another complete set of left coset representatives of H in G. Then,there exists P ∈Mat(m,RH) such that LT = P−1LT ′P.Example 3.1 Let G = Z/2Z = {0, 1}, H = {0}, and α = x0+ y1 ∈ RG. Then,we haveα(0 1) = (0 1)[x0 y0y0 x0].To obtain an expression for LT , we define the indicator function.χ by.χ(g) ={1 g ∈ H,0 g /∈ Hfor all g ∈ G.Lemma 3.2 Letα =∑g∈Gxgg.Then, we haveLT (α)ij =∑g∈G.χ(t−1i gtj)xgt−1i gtj.Proof of some properties of transfer 105Proof — We have(t1 t2 . . . tm)∑g∈G.χ(t−1i gtj)xgt−1i gtj16i6m,16j6m= m∑i=1∑g∈G.χ(t−1i gt1)xggt1 . . .m∑i=1∑g∈G.χ(t−1i gtm)xggtm=∑g∈Gxgg (t1 t2 . . . tm).This completes the proof. utFrom Lemma 3.2, we haveLT (g)ij =.χ(t−1i gtj)t−1i gtj=t−1i gtj t−1i gtj ∈ H,0 t−1i gtj 6∈ H.From t−1i gtj ∈ H if and only if gtj = ti, we haveLT (g)ij =(gtj)−1gtj t−1i gtj ∈ H,0 t−1i gtj 6∈ H.As for the definition of the noncommutative determinant, letψ : Mat(m,RH)→Mat (m,R (H/K))be an R-linear map such thatψ(hEij)= (hK)Eijfor all h ∈ H and 1 6 i, j 6 m, where Eij is the matrix with 1 in the (i j)entry and 0 otherwise. Obviously, ψ is an R-algebra homomorphism. Thedefinition of the noncommutative determinant is the following.Definition 3.3 We define the map Det : RG→ R (H/K) byDet = det ◦ψ ◦ LT .106 Naoya YamaguchiSince there is P such that LT = P−1LT ′P, we haveDet = det ◦ψ ◦ LT = det ◦ψ ◦ LT ′ .Thus, the determinant is invariant under a change of left regular represen-tations, so the determinant Det is well-defined. If K is the commutatorsubgroup of H, the determinant is similar to the Dieudonné determinant.Obviously, the map Det is a homomorphism. That is,Det(αβ) = Det(α)Det(β)for all α,β ∈ RG. Therefore, we obtain properties (1’) and (2’).Remark 3.4 In general, that α ∈ RG is invertible is not equivalent to thatDet(α) ∈ R(H/K)is invertible. For example, let R = C, Z/2Z ={0, 1}be the group of order two, S3be the symmetric group of degree three, G = Z/2Z× S3, H = S3, and K = [H,H]the commutator subgroup of H. Thenα =(0, e)+(0, (123))+(0, (132))is not invertible, where e is the unit element of H. But, Det(α) = 9K is invertible.4 Proof of the propertiesHere, we prove the transfer properties by using the noncommutative deter-minant’s properties.For all g ∈ G and for all t ∈ T , there exists a unique tj ∈ T such thatt−1i gtj ∈ H.Therefore, there exists sgn(g) ∈ {−1, 1} such thatDet(g) = det (ψ (LT (g)))= sgn(g)m∏i=1{(gti)−1gti}K = sgn(g)VG→H/K(g).Thus, we havesgn(gh)VG→H/K(gh) = Det(gh) = Det(g)Det(h)Proof of some properties of transfer 107= sgn(g)sgn(h)VG→H/K(g)VG→H/K(h).Hence, we obtainsgn(gh) = sgn(g)sgn(h),VG→H/K(gh) = VG→H/K(g)VG→H/K(h).Therefore, from property (1’) that Det is a homomorphism, the left transferVG→H/Kis a group homomorphism (assuming, that is, R = F2, and we do not con-sider the signature).Next, we show that the left transfer is invariant under a change of cosetrepresentatives by using property (2’) that the determinant is invariant undera change of regular representations. That is, we show thatm∏i=1{(gti)−1gti}K =m∏i=1{(gt ′i)−1gt ′i}Kwhere we define g = t ′i for all g ∈ t′iH.From property (2’), there exists sgn ′(g) ∈ {−1, 1} such thatm∏i=1{(gti)−1gti}K = sgn(g)Det(g)= sgn(g)sgn ′(g)m∏i=1{(gt ′i)−1gt ′i}K.Therefore, we have sgn(g)sgn ′(g) = 1 andm∏i=1{(gti)−1gti}K =m∏i=1{(gt ′i)−1gt ′i}K.Hence, the left transfer is invariant under a change of coset representatives.Now let us prove property (3) thatVG→H/K = ṼG→H/Kfrom property (3’) that any left regular representation is equivalent to anyright regular representation.LetG = Hu1 ∪Hu2 ∪ . . .∪Hum.108 Naoya YamaguchiThat is, we take a complete setU = {u1,u2, . . . ,um}of right coset representatives of H in G. Then, for all α ∈ RG, thereexistsRU(α) ∈Mat(m,RH)such that u1u2...umα = RU(α)u1u2...um .The R-algebra homomorphismRU : RG 3 α 7→ RU(α) ∈Mat(m,RH)is called the right regular representation.The same as the left transfer, we can show that the following lemma.Lemma 4.1 Let α =∑g∈G xgg. Then, we haveRU(α)ij =∑g∈G.χ(uigu−1j )xguigu−1j .Therefore, there exists s̃gn(g) ∈ {−1, 1} such that(det ◦ψ ◦ RU) (g) = s̃gn(g)ṼG→H/K(g)and ṼG→H/K is invariant under a change of coset representatives of H in G.We have properties (1) and (2).Since T is a complete set of left coset representatives of H in G, we cantake a complete set ofT−1 = {t−11 , t−12 , . . . , t−1m }of right coset representatives of H in G. Therefore,RT−1(α)ij =∑g∈G.χ(t−1i g(t−1j )−1)xgt−1i g(t−1j )−1= LT (α)ij.Proof of some properties of transfer 109We obtain property (3’). As a result,(det ◦ψ ◦ RU) (g) = (det ◦ψ ◦ LT ) (g).Therefore, we haves̃gn(g) = sgn(g),ṼG→H/K = VG→H/K.We obtain property (3). This completes the proof of Theorem 1.1.AcknowledgmentsI am deeply grateful to Prof. Tôru Umeda, Prof. Hiroyuki Ochiai, Prof. Mi-noru Itoh and Prof. Hideaki Morita who provided helpful comments andsuggestions. In particular, Tôru Umeda gave me the initial motivation forundertaking this study. I would also like to thank Cid Reyes for commentsand suggestions.REFERENCES[1] J.L. Alperin: “Sylow intersections and fusion”, J. Algebra 6 (1967),222–241.[2] E. Artin: “Geometric algebra”, Interscience, New York (1957).[3] H. Aslaksen: “Quaternionic determinants”, Math. Intelligencer 18(1996), 57–65.[4] D.C. Mayer: “Artin transfer patterns on descendant trees of fi-nite p-groups”, Adv. Pure Math. 6 (2016), 66–104.[5] M. Hall: “The Theory of Groups”, Macmillan, New York (1959).[6] B. Huppert: “Endliche Gruppen I”, Springer, Berlin (1967).[7] I. Schur: “Neuer Beweis eines Satzes über endliche Gruppen”, Sitzber.Akad. Wiss. Berlin (1902), 1013–1019.110 Naoya Yamaguchi[8] T. Umeda: “On some variants of induced representations”, Proc. Symp.Repr. Theory (2012), 7–17.Naoya YamaguchiCenter for Co-Evolutional Social SystemsKyushu University744 Motooka, Nishi-ku, Fukuoka 819-0395 (Japan)e-mail: n-yamaguchi@imi.kyushu-u.ac.jp

Proof of some properties of transfer using noncommutative determinants

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