Let G,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.
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 order p2q, 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 subgroup N 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]).
MNT 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]).
Let p and q be distinct primes and let G be a finite group of order p2q. Then one of the following holds:
p>q and G has a normal Sylow p-subgroup;
p<q and G has a normal Sylow q-subgroup;
p=2, q=3, G≅A4, and G has a normal 2-subgroup.
Proposition 4 (see [9]).
Let G be a nonabelian group of order p2q where p and q are distinct primes. Then exactly one of the following holds:
G′≅Zp and Gab≅Zpq;
G′≅Zp2 and Gab≅Zq;
G′≅Zp×Zp and Gab≅Zq;
G′≅Zq and Gab≅Zp2;
G′≅Zq and Gab≅Zp×Zp;
G′≅Z2×Z2.
Proposition 5 (see [12]).
The factor group G/G′ is abelian. If K is a normal subgroup of G such that G/K is abelian, then G′⊆K.
Proposition 6 (see [5]).
Let G≅Zm and H≅Zn be cyclic groups that act trivially on each other. Then G⊗H≅Z(m,n).
Proposition 7 (see [2]).
Let G be a finite group. Then
M(G) is a finite group whose elements have order dividing the order of G.
M(G)=1 if G is cyclic.
Proposition 8 (see [2]).
If the Sylow p-subgroups of G are cyclic for all p∣G, then M(G)=1.
Proposition 9 (see [2]).
Let N be a normal Hall subgroup of G and T a complement of N in G. Then
(3)MG≅MT×MNT.
Proposition 10 (see [2]).
If G1 and G2 are finite groups, then
(4)MG1×G2=MG1×MG2×G1⊗G2.
Proposition 11 (see [6]).
Let G 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]).
Let N=1; then M(G,N)=1.
Proposition 13 (see [3]).
Let N=G; then M(G,G)=M(G).
Proposition 14 (see [3]).
Suppose that G 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 order p2q are stated and proved. We assume that the group is nonabelian.
Theorem 15.
Let G 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.
Let G be a group of order p2q where p and q are distinct primes, and p<q. Since p<q, then by Proposition 3G 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.
Suppose N⊲G; then the Schur multiplier of pairs of G is computed below.
Case 1. If Gab≅Zp2 then by Proposition 11, M(G)=1.
Since M(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.
If N=1 then by Proposition 12, M(G,N)=M(G,1)=1.
If N=G then by Proposition 13, M(G,N)=M(G,G)=M(G). By Proposition 11, M(G)=Zp.
If N=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.
If N=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.
If N=Zpq, Zp2 or Zp×Zp then by similar way as in (iv), M(G,N)=Zp.
Theorem 16.
Let G 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.
Let G 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 case N=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).
If N=1 then M(G,N)=M(G,1)=1 (refer to Proposition 12).
If N=G then M(G,N)=M(G,G)=M(G) (refer to Proposition 13). Then, M(G)=Zp.
If N=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).
If N=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.
If N=Zp×Zp then by similar way as in (iv), M(G,N)=Zp.
If N=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 group G 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Ministry of Education (MOE), Malaysia, and Research Management Centre Universiti Teknologi Malaysia (RMCUTM) for the financial support through the Research University Grant (RUG) Vote no. 04H13. The second author would also like to thank Ministry of Education (MOE), Malaysia, for her MyPhD Scholarship.
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