# The Schur Multiplier of Pairs of Groups of Orderp2q

Authors: S. Rashid; A. A. Nawi; N. M. Mohd Ali; N. H. Sarmin
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

LetG,N be a pair of groups where G is a group and N is a normal subgroup of G. Then the Schur multiplier of pairs of groups G,N is a functorial abelian group MG,N. In this paper, MG,N for groups of order p2q where p and q are prime numbers are determined.


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## Body

## 1. Introduction

The Schur multiplier was introduced by Schur [1] in 1904. The Schur multiplier of a group G, M(G), is isomorphic to R∩[F,F]/[R,F] in which G is a group with a free presentation 1→R→F→G→1. He also computed M(G) for many different kinds of groups: for example, the dihedral group, metacyclic group, alternating group, and quaternion group. All computations of M(G) were then compiled by Karpilovsky [2] in a book entitled “The Schur Multiplier.”In 1998, Ellis [3] extended the notion of the Schur multiplier of a group to the Schur multiplier of a pair of group, (G,N), where N is a normal subgroup of G. The Schur multiplier of a pair of groups, (G,N), is a functorial abelian group M(G,N) whose principal feature is natural exact sequence
(1)H3G⟶ηH3GN⟶MG,N⟶MG⟶μMGN⟶NN,G⟶Gab⟶αGNab⟶1,
in which H3(-) denotes some finiteness-preserving functor from groups to abelian groups (to be precise, H3(-) is the third homology of a group with integer coefficients). The homomorphisms η, μ, α are those due to the functorial of H3(-), M(-), and (-)ab. Ellis [3] also stated that, for any pair (G,N) of groups, M(G,N)≅ker⁡(N∧G→G) where N∧G is the exterior product of N and G. The exterior product N∧G is obtained from N⊗G by imposing the additional relation n⊗g=1 for all (n,g)∈N∧G and the image of a general element n⊗g in N∧G is denoted by n∧g for all n∈N and g∈G.The nonabelian tensor product,G⊗H, was introduced by Brown and Loday [4] in 1987. G⊗H is the group generated by the symbols g⊗h subject to the relations
(2)gg′⊗h=gg′⊗hgg⊗h,g⊗hh′=g⊗hgh⊗hh′,
for all g, g′ in G and h, h′ in H. G⊗H is used in computing the Schur multiplier of the direct product of two groups, M(G×H). Some computations of the nonabelian tensor product of cyclic group of p-power order have been done by Visscher [5] in 1998.The nonabelian tensor square and Schur multiplier of groups of orderp2q, pq2, and p2qr has been computed by Jafari et al. [6]. In this paper, the Schur multiplier of pairs of groups of order p2q where p and q are primes is determined.In 2007, Moghaddam et al. [7] showed that M(G,N)≅R∩[S,F]/[R,F] if S is a normal subgroup of F such that N≅S/R. In 2012, Rashid et al. [8] determined the commutator subgroups of groups of order 8q. The Schur multiplier, nonabelian tensor square, and capability of groups of order p2q have been considered by Rashid et al. in [9], where p and q are distinct primes. In [10], they also computed the nonabelian tensor square and capability of groups of order 8q, where q is an odd prime.


## 2. Preliminaries

This section includes some preliminary results that are used in proving our main theorems.Definition 1 (see [2]).
A normal subgroupN of G is called a normal Hall subgroup of G if the order of N is coprime to its index in G.Definition 2 (see [2]).
M
N
T is defined as the T-stable subgroup of M(N); that is, M(N)T={f∈M(N)ConNt(f)=fforallt∈T} where T is a subgroup of G in which G is the semidirect product of a normal subgroup N and a subgroup T, and ConNt(f) is the conjugation of t on f.Proposition 3 (see [11]).
Letp and q be distinct primes and let G be a finite group of order p2q. Then one of the following holds:(i)
p>q and G has a normal Sylow p-subgroup;(ii)
p<q and G has a normal Sylow q-subgroup;(iii)
p=2, q=3, G≅A4, and G has a normal 2-subgroup.Proposition 4 (see [9]).
LetG be a nonabelian group of order p2q where p and q are distinct primes. Then exactly one of the following holds:(i)
G′≅Zp and Gab≅Zpq;(ii)
G′≅Zp2 and Gab≅Zq;(iii)
G′≅Zp×Zp and Gab≅Zq;(iv)
G′≅Zq and Gab≅Zp2;(v)
G′≅Zq and Gab≅Zp×Zp;(vi)
G′≅Z2×Z2.Proposition 5 (see [12]).
The factor groupG/G′ is abelian. If K is a normal subgroup of G such that G/K is abelian, then G′⊆K.Proposition 6 (see [5]).
LetG≅Zm and H≅Zn be cyclic groups that act trivially on each other. Then G⊗H≅Z(m,n).Proposition 7 (see [2]).
LetG be a finite group. Then (i)
M(G) is a finite group whose elements have order dividing the order of G.(ii)
M(G)=1 if G is cyclic.Proposition 8 (see [2]).
If the Sylowp-subgroups of G are cyclic for all p∣G, then M(G)=1.Proposition 9 (see [2]).
LetN be a normal Hall subgroup of G and T a complement of N in G. Then
(3)MG≅MT×MNT.Proposition 10 (see [2]).
IfG1 and G2 are finite groups, then
(4)MG1×G2=MG1×MG2×G1⊗G2.Proposition 11 (see [6]).
LetG be a finite nonabelian group. If G is a group of order p2q, then
(5)MG=1,ifG′=Zq,Gab=Zp2,Zp,ifG′=Zq,Gab=Zp×Zp,Z2,ifGab=Z2×Z2.The following propositions are some of the basic results of the Schur multiplier of a pair deduced by Ellis [3], assuming only the existence of the natural exact sequence in (1) and the existence of a certain transfer homomorphism.Proposition 12 (see [3]).
LetN=1; then M(G,N)=1.Proposition 13 (see [3]).
LetN=G; then M(G,G)=M(G).Proposition 14 (see [3]).
Suppose thatG is a finite group. Let the order of the normal subgroup N be coprime to its index in G and T a complement of N in G. Then G≅N⋊T and M(G,N)≅M(N)T.


## 3. Main Result

In the following two theorems, the Schur multipliers of pairs of groups of orderp2q are stated and proved. We assume that the group is nonabelian.Theorem 15.
LetG be a group of order p2q where p and q are distinct primes, and p<q. If N⊲G, then the Schur multiplier of pairs of G(6)MG,N=1,ifGab≅Zp2orGab≅Zp×ZpwhenN=1orZq,Zp,ifGab≅Zp×ZpwhenN=G,Zp,Zpq,Zp×ZporZp2,
where Gab=G/G′.Proof.
LetG be a group of order p2q where p and q are distinct primes, and p<q. Since p<q, then by Proposition 3  G has a normal Sylow q-subgroup: call it Q. Moreover, [G:Q]=p2 so G/Q is abelian. Then by Proposition 5, we have G′⊆Q; that is, G′=Zq. Thus, by Proposition 4, Gab≅Zp2or Gab≅Zp×Zp.
SupposeN⊲G; then the Schur multiplier of pairs of G is computed below.
Case
1. If Gab≅Zp2 then by Proposition 11, M(G)=1.
SinceM(G)=1, for all normal subgroups N of G, M(G,N)≤M(G)=1.
Case
2. If Gab≅Zp×Zp then by Proposition 11, M(G)=Zp.(i)
IfN=1 then by Proposition 12, M(G,N)=M(G,1)=1.(ii)
IfN=G then by Proposition 13, M(G,N)=M(G,G)=M(G). By Proposition 11, M(G)=Zp.(iii)
IfN=Zq then G is the semidirect product of Zq and H in which N and [G:N] are coprimes, and N is a normal Hall subgroup of G (refer to Definition 1). Therefore by Proposition 14, M(G,N)=M(N)H=1 since M(Zq)=1 (refer to Proposition 7). Note that, for this case, G/N=G/G′≠Zp2; that is, G/N≠Zp2.(iv)
IfN=Zp then G/N is nonabelian group of order pq. (If G/N≅Zpq then by Proposition 5, G′⊆N; that is, Zq⊆Zp and this statement is a contradiction). Thus the exact sequence M(G,N)→M(G)→M(G/N)=1 shows that M(G,N)/κ≅Zp where κ is the kernel of homomorphism M(G,N) to M(G). Then M(G,N)=Zp.(v)
IfN=Zpq, Zp2 or Zp×Zp then by similar way as in (iv), M(G,N)=Zp.Theorem 16.
LetG be a group of order p2q where p and q are distinct primes, and p>q. If N⊲G, then the Schur multiplier of pairs of G(7)MG,N=1,ifG′≅ZporG′≅Zp2,orG′≅Zp×ZpwhenN=1orZq,Zp,ifG′≅Zp×ZpwhenN=G,ZporZp×Zp,
where Gab=G/G′.Proof.
LetG be a group of order p2q where p and q are distinct primes, and p>q. Since p>q, G has a normal Sylow p-subgroup, namely, P (refer to Proposition 3). [G:P]=q so G/P is abelian. Hence, G′⊆P (refer to Proposition 5); that is, G′≅Zp×Zp, Zp2 or Zp. Suppose N⊲G; then the Schur multiplier of pairs of G is computed below.
(In this caseN=1,Zq,Zp,Zp×Zp and G.)
Case
1. If G′≅Zp×Zp then G′=p2 and [G:G′]=q are coprimes. Then, by Definition 1, G′ is a normal Hall subgroup of G. Therefore by Proposition 9, M(G)=M(T)×M(G′)T where T is a complement of G′ and T≅Zq. Thus, M(G)=M(T)×M(Zp×Zp)T.M(T)=1 (refer to Proposition 7). Hence, M(G)=Zp (refer to Propositions 10, 6, and 7). (i)
IfN=1 then M(G,N)=M(G,1)=1 (refer to Proposition 12).(ii)
IfN=G then M(G,N)=M(G,G)=M(G) (refer to Proposition 13). Then, M(G)=Zp.(iii)
IfN=Zq then N is a normal Hall subgroup of G (refer to Definition 1) and G is the semidirect product of N and H in which H is a complement of N in G. Therefore by Proposition 14, M(G,N)=M(N)H=1 since M(Zq)=1 (refer to Proposition 7).(iv)
IfN=Zp then G/N is nonabelian group of order pq. (If G/N≅Zpq then by Proposition 5, G′⊆N; that is, Zp×Zp⊆Zp and this statement is a contradiction). Thus the exact sequence M(G,N)→M(G)→M(G/N)=1 shows that M(G,N)/κ≅Zp where κ is the kernel of homomorphism M(G,N) to M(G). Then M(G,N)=Zp.(v)
IfN=Zp×Zp then by similar way as in (iv), M(G,N)=Zp.(vi)
IfN=Zpq or Zp2 then G/N is abelian group and G′≅Zp×Zp⊆N≅Zpq or Zp2 but this statement is a contradiction. So M(G,N) when N=Zpq or Zp2 are not considered.
Case
2. If G′≅Zp2 then G/G′≅Gab≅Zq. Hence, all Sylow subgroups of G are cyclic. Therefore, by Proposition 8, M(G)=1. Thus, for all normal subgroups N of G, M(G,N)≤M(G)=1.
Case
3. If G′≅Zp then M(G)=1 since M(G)=M(G′)×M(K) where K is a group of order pq. By Propositions 7 and 8, M(G′)=1 and M(K)=1. Thus, for all normal subgroups N of G, M(G,N)≤M(G)=1.


## 4. Conclusion

For a groupG of order p2q where p and q are prime numbers, Q is the unique normal Sylow q-subgroups of G if p<q, while P is the unique normal Sylow p-subgroups of G if p>q. In this paper, we determined the Schur multiplier of pairs of groups of order p2q. Our proofs show that MG,N for groups of order p2q is either 1 or Zp.



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