Pradeep K. Rai

Comptes Rendus Mathématique

Comptes RendusMathématiquePradeep K. RaiOn the occurrence of elementary abelian p-groups as the Schur multiplier ofnon-abelian p-groupsVolume 361 (2023), p. 803-806Published online: 11 May 2023https://doi.org/10.5802/crmath.445This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orge-ISSN : 1778-3569Comptes RendusMathématique2023, Vol. 361, p. 803-806https://doi.org/10.5802/crmath.445Group theory / Théorie des groupesOn the occurrence of elementary abelianp-groups as the Schur multiplier ofnon-abelian p-groupsPradeep K. Raiaa Mahindra University, Hyderabad, Telangana, IndiaE-mail: pradeepkumar.rai@mahindrauniversity.edu.inAbstract. We prove that every elementary abelian p-group, for odd primes p, occurs as the Schur multiplierof some non-abelian finite p-group.Keywords. Schur multiplier, finite p-group.Mathematical subject classification (2010). 20J99, 20D15.Funding. This research is supported by INSPIRE Faculty Research Grant DST/INSPIRE/04/2016/001594.Manuscript received 25 August 2022, revised and accepted 21 November 2022.1. IntroductionThe Schur multiplier M(G) of a finite group G is defined as the second cohomology group of Gwith coefficients in C∗. It plays an important role in the theory of extensions of groups. Findingbounds on the order, exponents, and ranks of the Schur multiplier of prime power groups, as wellas classifying groups with trivial Schur multipliers, have been main lines of investigation in thepast. Another problem where it appears that not much progress has been made is the following:Question 1. Is it true that every finite abelian p-group is isomorphic to the Schur multiplier ofsome non-abelian finite p-group? [5, Question 15.30].More generally,Question 2. Which abelian groups occur as the Schur multipliers of non-abelian finitegroups? [6].The immediate questions that arise from these questions are the following:Question 3. For all natural numbers n does there exist a non-abelian p-group G such that|M(G)| = pn?Question 4. For all natural numbers e does there exist a non-abelian p-group G such thatexponent of M(G) is pe ?ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/804 Pradeep K. RaiQuestion 5. For all natural numbers d does there exist a non-abelian p-group G such that theminimal number of generators of M(G) is d?It is easy to see that the answers to the Questions 3 and 4 are affirmative. To see this, considerG = Ep × (Zpm+1 ×Zpm ),where Ep is the extra-special p-group of order p3 and exponent p, and Zn denote the cyclicgroup of order n. For a group G , let γ2(G) denote the derived subgroup of G . Applying [4,Theorem 2.2.10] we get thatM(G) ∼= M(Ep )×M(Zpm+1 ×Zpm )×(Epγ2(Ep )⊗ (Zpm+1 ×Zpm )).Therefore |M(G)| = p2pm p4 = pm+6 and exponent of M(G) is pm . See [2, 3] for the existence ofnon-abelian groups having Schur multiplier of order pn for each n ≤ 6.We now state the main result of this article, which gives an affirmative answer to the Ques-tion 5:Theorem 6. Given a natural number n and an odd prime p, there exists a non-abelian p-groupwhose Schur multiplier is elementary abelian of order pn .2. ProofBefore proceeding to the proof of Theorem 6, we set some notations that are mostly standard.Let G be a group and x, y ∈G . Then [x, y] denotes the commutator x−1 y−1x y . By d(G) we denotethe minimal number of generators of G . The exponent of the group G is denoted by exp(G). Wewrite γi (G) for the i -th term in the lower central series of G . By Zpn we denote the cyclic group oforder pn .The proof is founded on the following two results of Blackburn and Evens [1].Proposition 7 ([4, Corollary 3.2.4]). Let G be a finite nilpotent group of class c ≥ 2 and let G = F /Rbe a presentation of G as a factor group of a free group of finite rank. Then(1) upon identifying G/γ2(G) with F /γ2(F )R and γc (G) with γc (F )R/R, the mapG/γ2(G)⊗γc (G) λ7−→ γc+1(F )/([F,R]∩γc+1(F ))f γ2(F )R ⊗xR 7−→ [ f , x]([F,R]∩γc+1(F ))is a surjective homomorphism.(2) Let X = Kerλ. Then there is an exact sequence1 7−→ X 7−→G/γ2(G)⊗γc (G) 7−→ M(G) 7−→ M(G/γc (G)) 7−→ γc (G) 7−→ 1.Lemma 8 ([1, Remark, Section 3]). Let G be a group of nilpotency class 2 with G/γ2(G) elementaryabelian. Then X , as defined in Proposition 7, is generated by all elements of the form xγ2(G)⊗[y, z]+yγ2(G)⊗ [z, x]+ zγ2(G)⊗ [x, y] and all elements of the form xγ2(G)⊗xp .The proof of the Theorem 6 also makes use of the following lemma. The proof of the lemmacan be extracted from the proof of [7, Proposition 3.2].Lemma 9. Let G be a special p-group minimally generated by x1, x2, . . . , xd , and WG be thesubgroup of G/γ2(G)⊗γ2(G) generated by all elements of the form xγ2(G)⊗[y, z]+ yγ2(G)⊗[z, x]+zγ2(G)⊗ [x, y], x, y, z ∈G. Then WG is generated by BG whereBG = {xiγ2(G)⊗ [x j , xk ]+x jγ2(G)⊗ [xk , xi ]+xkγ2(G)⊗ [xi , x j ] | 1 ≤ i < j < k ≤ d}.Pradeep K. Rai 805For a special p-group G notice that G/γ2(G) and γ2(G) are elementary abelian and thereforecan be regraded as vector spaces over Zp . With this in mind, we are now ready to present thefollowing two propositions, that are the main ingredients of the proof of the Theorem 6.Proposition 10. For an odd prime p and natural numbers d and j , where d ≥ 4 and 3 ≤j +2 ≤ d, let G j be the special p-group of exponent p minimally generated by x1, x2, . . . , xd , suchthat all commutators [xr , xs ], other than [x1, x2], [x2, x3], . . . [x j+1, x j+2], are identity and the set{[x1, x2], [x2, x3], . . . [x j+1, x j+2]} is linearly independent. Then M(G j ) is elementary abelian of orderp12 d(d−1)+2 j+1.Proof. Since G j is a group of nilpotency class 2, exp(M(G j )) divides exp(G). Therefore M(G j ) isan elementary abelian group. Now, using the exact sequence given in Proposition 7, we get that∣∣M(G j )∣∣=∣∣∣M( G jγ2(G j )| )∣∣∣∣∣γ2(G j )∣∣∣∣∣ G jγ2(G j ) ⊗γ2(G j )∣∣∣∣∣X ∣∣ . (1)Let WG j and BG j be the sets defined in Lemma 9. To obtain |X |, note from Lemma 8 that|X | = |WG j | because the group G j is of exponent p. Let (x, y, z) denotes the element xγ2(G j )⊗[y, z]+yγ2(G j )⊗[z, x]+zγ2(G j )⊗[x, y] ∈G j /γ2(G j )⊗γ2(G j ) for x, y, z ∈G j . Consider the followingsubset DG j of BG jDG j ={(xi , xi+1, xk )∣∣ i = 1,2, . . . j +1, k = 1,2, . . .d and k ̸= i −1, i , i +1},and notice that all the elements of BG j can be generated by elements of DG j . As a result DG jgenerates WG j . We further claim that DG j is a basis for WG j . To see this, letj+1∑i=1d∑k=1k ̸=i−1,i ,i+1αi k (xi , xi+1, xk ) = 0.This givesj+1∑i=1(( d∑k=1k ̸=i−1,i ,i+1αi k xk)+α(i−1)(i+1)xi−1)⊗ [xi , xi+1] = 0.Now, since {x1γ2(G), x2γ2(G), . . . xdγ2(G)} and {[x1, x2], [x2, x3], . . . , [x j+1, x j+2]} are linearly inde-pendent, it follows that αi k = 0 for all i = 1,2, . . . j +1, k = 1,2, . . .d ,k ̸= i −1, i , i +1. Next, noticethat |DG j | = (d − 3) j + 1, and hence |WG j | = p(d−3) j+1. From [4, Corollary 2.2.12] we also havethat |M(G/γ2(G j ))| = p 12 d(d−1), because G j /γ2(G j ) is an elementary abelian p-group of order pd .Putting these values and |γ2(G j )| = p j+1 in Equation (2.1) we get that|M(G j )| = p12 d(d−1)+2 j+1. □The following Proposition can be proved along the same lines.Proposition 11. For an odd prime p and a natural number j , where 4 ≤ j + 3 ≤ d, let G jbe the special p-group of exponent p minimally generated by x1, x2, . . . , xd , d ≥ 4, such that allcommutators [xr , xs ], other than [x1, x2], [x2, x3], . . . [x j , x j+1], [x j+2, x j+3], are identity and theset {[x1, x2], [x2, x3], . . . [x j , x j+1], [x j+2, x j+3]} is linearly independent. Then M(G j ) is elementaryabelian of order p12 d(d−1)+2 j .We are now ready to prove the Theorem 6.Proof of Theorem 6. Let m be a natural number such that m ≥ 10. Then there exist unique non-negative integers k ≥ 5 and l ≤ k −1 such that m = 12 k(k −1)+ l . If l is a non-zero even number,i.e., l = 2 j for some j ≥ 1, consider the group G j constructed in Proposition 11 with d = k sothat |M(G j )| = pm . In case l is an odd number greater than or equal to 3, i.e., l = 2 j + 1 for806 Pradeep K. Raisome j ≥ 1, consider the group G j constructed in Proposition 10 with d = k. If l = 0 and k isan even number, consider the group G j constructed in 10 with d = k − 1 and j = k2 − 1, so that|M(G j )| = p 12 (k−1)(k−2)+k−1 = p 12 k(k−1). If l = 0 and k is an odd number, consider the group G jconstructed in 11 with d = k−1 and j = k−12 , so that |M(G j )| = p12 (k−1)(k−2)+k−1 = p 12 k(k−1). If l = 1and k is an even number, consider the group G j constructed in 11 with d = k −1 and j = k2 , sothat |M(G j )| = p 12 (k−1)(k−2)+k = p 12 k(k−1)+1. If l = 1 and k is an odd number, consider the group G jconstructed in 10 with d = k−1 and j = k−12 , so that |M(G j )| = p12 (k−1)(k−2)+k−1+1 = p 12 k(k−1)+1. Forthe existence of groups with elementary abelian Schur multiplier of order pm ,m ≤ 9, see [2,3]. □References[1] N. Blackburn, L. Evens, “Schur multipliers of p-groups”, J. Reine Angew. Math. 309 (1979), p. 100-113.[2] B. Eick, T. J. Ghorbanzadeh, “Computing the Schur multipliers of the Lie p-rings in the family defined by a symbolicLie p-ring presentation”, J. Symb. Comput. 106 (2021), p. 68-77.[3] S. Hatui, V. Kakkar, M. K. Yadav, “The Schur multiplier of groups of order p5”, J. Group Theory 22 (2019), no. 4, p. 647-687.[4] G. Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Series, vol. 2, Clarendon Press,1987.[5] E. I. Khukhro, V. D. Mazurov (eds.), The Kourovka notebook. Unsolved problems in group theory, 20th ed., SobolevInstitute of Mathematics, 2022.[6] P. Moravec, “A talk in Groups St Andrews”, 2009, https://mathshistory.st-andrews.ac.uk/Groups/2009/slides/moravec.pdf.[7] P. K. Rai, “On the Schur multiplier of the Special p-groups”, J. Pure Appl. Algebra 222 (2018), no. 2, p. 316-322.

On the occurrence of elementary abelian $p$-groups as the Schur multiplier of non-abelian $p$-groups

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On the occurrence of elementary abelian $p$ -groups as the Schur multiplier of non-abelian $p$ -groups

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On the occurrence of elementary abelian ppp-groups as the Schur multiplier of non-abelian ppp-groups

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On the occurrence of elementary abelian $p$ -groups as the Schur multiplier of non-abelian $p$ -groups