www.theoryofgroups.ir
ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 3 No. 1 (2014), pp. 65-72.
c
2014 University of Isfahan
www.ui.ac.ir
MAXIMAL SUBSETS OF PAIRWISE NON-COMMUTING ELEMENTS OF
p-GROUPS OF ORDER LESS THAN p6
R. ORFI
Communicated by Alireza Abdollahi
Abstract. LetGbe a non-abelian group of order pn, wheren≤5 in whichGis not extra special of order p5. In this paper we determine the maximal size of subsetsX of Gwith the property that
xy6=yxfor anyx, yinX withx6=y.
1. Introduction
Let Gbe a non-abelian group. A subset X of Gis said to be a subset of pairwise non-commuting
elements of G ifxy 6=yx for any x, y in X with x 6=y. The maximal cardinality of these subsets is denoted byω(G). Alsoω(G) is the maximal clique size in the non-commuting graph of a finite group G. LetZ(G) be the center ofG. The non-commuting graph of a groupGis a graph withG\Z(G) as the set of vertices and two vertices are joined if and only if they do not commute. By a famous result
of B. H. Neumann [11], answering a question of P. Erd¨os, the finiteness of ω(G) in Gis equivalent to the finiteness of the factor group G/Z(G). Furthermore many attempts have been made to findω(G) for some groups G, see for example [1], [2], [3], [5], [6], [7], [11] [12]. In this paper we find ω(G) for all non-abelian p-groups G with |G| ≤p5 in which G is not extra special of order p5. To prove this we use the classification of p-groups by James [10], which based on isoclinism. Following [4, §29], two groups are isoclinic if their commutator subgroups and central quotients are isomorphic and their
commutator operations are essentially the same.
Throughout this paper the following notation is used. All groups are assumed to be finite. The
letter p denotes a prime number. CG(x) is the centralizer of an element x in a group G. The terms
MSC(2010): Primary: 20D15; Secondary: 20D60.
keywords: p-group, AC-group, Pairwise non-commuting elements. Received: 15 July 2013, Accepted: 16 September 2013.
of the lower central series of Gare denoted byγi=γi(G). The minimal number of generators of Gis denoted byd(G). We write [a, b] for a−1b−1ab. AlsoZnis the cyclic group of ordernand (Zn)kis the direct product of k copies ofZn. We write SmallGroup(n,m) for the mth group of order n as quoted
in the “Small Groups” library in GAP[8]. All unexplained notation is standard and follows that of
[4].
2. Some Basic Results
In this section we give some basic results that are needed for the main results of the paper. A group
Gis called anAC-group if the centralizer of every non-central element ofGis abelian. In this section first we give some results for AC-groupG and findω(G) in some cases.
Lemma 2.1. [13, Lemma 3.2 ]The following are equivalent on a groupG.
(i) Gis an AC-group.
(ii) If [x, y] = 1, then CG(x) =CG(y), where x, y∈G\Z(G).
Lemma 2.2. Let G be anAC-group.
(i) If a, b∈G\Z(G) with distinct centralizers, thenCG(a)∩ CG(b) =Z(G).
(ii) If G=∪k
i=1CG(ai), where CG(ai) and CG(aj) are distinct for 1≤i < j ≤k, then{a1, . . . , ak}
is a set of pairwise non-commuting elements in Gof maximal cardinality.
Proof. This is straightforward. See also [6, Lemma 2.2 ].
Lemma 2.3. [6, Lemma 2.3 ]LetGbe a group of order pn with the central quotient of orderp2, where
p is a prime number. Then G is an AC-group andω(G) =p+ 1.
Lemma 2.4. Let G be a group of order pn with the central quotient of order p3, where p is a prime number. Then
(i) Gis an AC-group.
(ii) If G possesses no abelian maximal subgroup, thenω(G) =p2+p+ 1.
(iii) IfG possesses an abelian maximal subgroup, thenGhas exactly one abelian maximal subgroup and ω(G) =p2+ 1.
Proof. See [3, Lemma 3.2(i) and Theorem 3.3 ].
Lemma 2.5. Let G=A×B, where A is an abelian subgroup of G. Then
ω(G) =ω(B).
Proof. This is clear.
Now following [4, §29], we define isoclinism between two groups and we state that isoclinic groups
and H are said to be isoclinic provided there exist two isomorphisms f : G/Z(G) → H/Z(H) and f0 :G0→H0 such that iff(g1Z(G)) =h1Z(H) andf(g2Z(G)) =h2Z(H), thenf0([g1, g2]) = [h1, h2].
In this case we writeG∼H.
Lemma 2.6. Let G∼H. Then
(i) Gis an AC-group if and only if H is an AC-group,
(ii) ω(G) =ω(H).
Proof. (i) Let G be an AC-group. We claim that CH(h) is abelian for any h ∈ H\Z(H). Assume that h1, h2 ∈ CH(h). By the above definition we see that there exist g, g1 and g2 in G, such that
f(gZ(G)) =hZ(H),f(g1Z(G)) =h1Z(H) and f(g2Z(G)) =h2Z(H). Moreoverf0([g, gi]) = [h, hi] = 1 for 1≤i≤2. This implies that g1, g2∈ CG(g), which completes the proof.
(ii) LetX={a1, . . . , an}be a subset of pairwise non-commuting elements ofGof maximal cardinality. Therefore aiZ(G) 6= ajZ(G) for 1 ≤ i < j ≤ n. This implies that f(aiZ(G)) 6= f(ajZ(G)) . Let
f(aiZ(G)) = biZ(H) for 1 ≤ i ≤ n. Therefore {b1, . . . , bn} is a subset of pairwise non-commuting elements of H since 1 6= f0([ai, aj]) = [bi, bj]. This yields that ω(G) ≤ ω(H). Now by the same
argument we see thatω(H)≤ω(G), as desired.
For the rest of this section we give some results forp-groups of maximal class. LetGbe ap-group of maximal class and orderpn (n≥4), wherepis a prime. Following [9], we define the 2-step centralizer Ki in G to be the centralizer in G of γi(G)/γi+2(G) for 2 ≤ i ≤ n−2 and define Pi = Pi(G) by P0 =G,P1 =K2,Pi =γi(G) for 2≤i≤n. The degree of commutativity l=l(G) ofG is defined to be the maximum integer such that [Pi, Pj]≤Pi+j+l for alli, j≥1 if P1 is not abelian andl=n−3
ifP1 is abelian.
Lemma 2.7. LetGbe ap-group of maximal class which possesses an abelian maximal subgroup. Then
P1 is abelian.
Proof. See [6, Lemma 3.1].
Theorem 2.8. [9, Theorem 3.2.11 ] LetG be ap-group of maximal class and order pn where nis odd
and 5≤n≤2p+ 1, then Ghas positive degree of commutativity.
Corollary 2.9. Let G be a p-group of maximal class and order p5.
(i) If G possesses an abelian maximal subgroup, thenω(G) =p3+ 1.
(ii) If G possesses no abelian maximal subgroup, thenω(G) =p3+p+ 1.
Proof. By Theorem 2.8, we see that [P1, P3] = 1. Therefore the result follows from [6, Theorems 3.4
3. Main Results
In this section we determine ω(G) for all non-abelian groups G of order pn, where n≤5 in which
Gis not an extra special group of orderp5. Forn= 3, we see thatω(G) =p+ 1 by Lemma 2.3. First forp= 2 we use the following program in GAP, which computes ω(G):
LoadPackage("grape");
N:=function(a,b)
return(IsAbelian(Group(a,b)));
end;
NonCommutingGraph:=function(g)
local k, x, y;
k:=Graph(g,Difference(g,Center(g)),OnPoints,function(x,y) return N(x,y)=false;end);
return k;
end;
clique:=function(x)
local G1,G2;
G1:=NonCommutingGraph(x);
G2:=ComplementGraph(G1);
return Size(IndependentSet(G2));
end;
CliqueNumber:=function(x)
local c, t, M;
c:=clique(x);
while c>0 do
t:=c;
M:=CompleteSubgraphsOfGivenSize(NonCommutingGraph(x),c+1,0);
c:=Size(M);
if c=0 then return(t); fi;
od;
end;
Lemma 3.1. Let G be a non-abelian group of order24
(i) If G is of maximal class, thenω(G) = 5.
(ii) If G is of class two, then ω(G) = 3.
Lemma 3.2. LetGbe a non-abelian group of order25. ThenG= SmallGroup(25, i), where1≤i≤51
(i) ω(G) = 3, where i∈ {2,4,5,12,17,22,23,24,25,26,37,38,46,47,48} .
(ii) ω(G) = 5, where i∈ {9,10,11,13,14,15,27,28,29,30,31,32,33,34, 35,39,40,41,42,49,50}.
(iii) ω(G) = 6, where i∈ {6,7,8,43,44}. (iv) ω(G) = 9, where i∈ {18,19,20}.
For the rest of the paper we assume thatp >2.Now by [10], all groups of orderpn, wheren∈ {4,5} are classified in isoclinic families. In fact we see that there are two non-isoclinic families of non-abelian
groups of order p4 and by Lemma 2.6, it is enough to select one group Gfrom each family and find
ω(G). By the notation of [10], we selectH1 = Φ2(14) and H2 = Φ3(14), which are non-isoclinic groups
of order p4. Similarly by [10], we have eight non-isoclinic families of non-abelian groups of order p5, which are not extra special. So we select one group Gi from each non-isoclinic families for 1≤i≤8.
By using the notation of [10], we select Gi as below:
G1 = Φ2(311)a, G2 = Φ3(15), G3= Φ4(221)a, G4 = Φ6(15),
G5 = Φ7(15), G6 = Φ8(32), G7 = Φ9(2111)a G8= Φ10(15).
Moreover following [10],αi(p+1) will denote the wordαpi+1α(
p
2) i+2. . . α
(p k)
i+k. . . αi+pwhereiis a positive integer and αi+2. . . αi+p are suitably defined. For economy of space, all relations of the form [α, β] = 1 (with α, β generators) have been omitted from the any given presentation and should be assumed when reading it. By using Table 4.1 of [10], we list the following properties of groups H1, H2 and Gi for 1 ≤i ≤8 in the table below. Moreover ω(G) is listed bellow for these groups and the proofs are in the following lemmas.
Table 1
Group |G/Z(G)| G0 cl(G) ω(G)
H1 p2 Zp 2 1 +p
H2 p3 (Zp)2 3 1 +p2
G1 p2 Zp 2 1 +p
G2 p3 (Zp)2 3 1 +p2
G3 p3 (Zp)2 2 1 +p2
G4 p3 (Zp)3 3 p2+p+ 1
G5 p4 (Zp)2 3 p(p+ 1)
G6 p4 Zp2 3 p(p+ 1)
G7 p4 (Zp)3 4 p3+ 1
G8 p4 (Zp)3 4 p3+p+ 1
Lemma 3.3. We haveω(H1) = 1 +p and ω(H2) = 1 +p2.
Proof. By Lemma 2.3, we haveω(H1) = 1 +psince|H1/Z(H1)|=p2. Moreover by using Table 1, we
have H20 ∼= (Zp)2 and|Z(H2)|=pand so we see thatH2/CH2(H
0
2),→GL(2, p). Hence we deduce that
CH2(H
0
2) is an abelian subgroup of orderp3 inH2. Therefore Lemma 2.4(iii) completes the proof.
(i) If G is of maximal class, thenω(G) = 1 +p2.
(ii) If G is of class two, then ω(G) = 1 +p.
Proof. We see that G∼H2 when G is of maximal class and G∼H1 when G is of class two by [10,
4.4]. Now the result follows from Lemma 2.6(ii).
Lemma 3.5. We have
(i) ω(G1) = 1 +p,
(ii) ω(G2) = 1 +p2,
(iii) ω(G3) = 1 +p2.
Proof. (i) This is evident from Lemma 2.3 and Table 1.
(ii) By [10, 4.4 and 4.5] we may write G2 = B ×Zp, where B is of maximal class and order p4. Thereforeω(B) = 1 +p2 by Corollary 3.4(i). Hence the result follows from Lemma 2.5.
(iii) By [10, 4.5], we have the following presentation for G3 :
G3 = hα, α1, α2, β1, β2|[αi, α] = βi, αp = β2, α1p = β1, α2p = βip = 1 (i = 1,2)i . On Setting H =hα1, α2, β1, β2i, we see that H is an abelian subgroup of order p4. Hence we may conclude the
result by Lemma 2.4(iii) and Table 1.
Lemma 3.6. We haveω(G4) =p2+p+ 1.
Proof. By Table 1, we deduce that the Frattini subgroup of G4 is equal to G04 and so d(G4) = 2.
ThereforeG4 has 1 +p distinct maximal subgroups. By [10, 4.5], G4 has the following presentation:
G4 =hα1, α2, β, β1, β2|[α1, α2] =β,[β, αi] =βi, αpi =β p=βp
i = 1 (i= 1,2)i.
On settingM0 =hα2, β, β1, β2iand Mi=hα1αi2, β, β1, β2ifor 1≤i≤p. It is easy to check that these
non-abelian maximal subgroups are distinct. Therefore the result follows from Lemma 2.4(ii).
Lemma 3.7. We haveω(G5) =p(p+ 1).
Proof. By [10, 4.5],G5 has the following presentation :
G5 =hα, α1, α2, α3, β|[αi, α] =αi+1,[α1, β] =α3, αp =α(1p)=α
p
i+1 =βp = 1 (i= 1,2)i.For p= 3, by
using GAP we see thatG5 = SmallGroup(243,58) andω(G5) = 12. For p >3, we haveα(1p)=α
p
1 = 1
and by using Table 1, we see that G05 =hα2, α3i,CG5(G
0
5) = hα1, α2, α3, βi and Z(G5) = hα3i. Now
on setting M0 = hα1, α2, α3, βi and Mi = hααi1, α2, α3, βi for 1 ≤ i ≤ p, it is easy to check that
these non-abelian maximal subgroups are distinct and Mi ∩Mj = K for 0 ≤ i 6= j ≤ p, where K =hα2, α3, βi and soG=Spi=0Mi. ObviouslyZ(M0) =hα2, α3i ,Z(Mi) =hαi2β, α3i for 1≤i≤p
and Z(Mi)∩Z(Mj) = Z(G) for 0 ≤ i 6= j ≤ p. Hence K = Spi=0Z(Mi). By Lemma 2.3, we have ω(Mi) = 1 + p. Let T = {x1, . . . , xm} be a subset of pairwise non-commuting elements in G5. We claim that |T ∩(Mi \K)| ≤ p, for otherwise suppose that x1, . . . , x1+p ∈ Mi\K. Hence Mi =Spj=1+1CMi(xj) sinceω(Mi) = 1 +p. By the fact thatxj ∈/ K we haveZ(Mi)≤ CMi(xj)∩K K
and soZ(Mi) =CMi(xj)∩K for 1≤j≤1 +p. ThereforeK=K∩Mi=
Sp+1
j=1(CMi(xj)∩K) =Z(Mi),
sinceK =Sp
i=0Z(Mi) and if|T∩K|= 0 we deduce that m≤p(p+ 1). Therefore ω(G5)≤p(p+ 1).
On settingA={αα1jβi, α1βi, ααi2β} for 0≤i≤p−1 and 1≤j≤p−1 it is easy to check that A is
a subset of pairwise non-commuting elements ofG5 of orderp(p+ 1), which completes the proof.
Lemma 3.8. We have
(i) ω(G6) =p(p+ 1),
(ii) ω(G7) =p3+ 1,
(iii) ω(G8) =p3+p+ 1.
Proof. (i) By [10, 4.5], we have the following presentation forG6:
G6 =hα1, α2, β|[α1, α2] =β =α1p, βp
2 =α2p
2 = 1i.
Therefore G6 = hα1, α2i, hα1iEG6 and G6/hα1i is cyclic. The rest follows from Table 1 and [7,
Theorem 1.1], .
(ii) By Table 1,G7 is of maximal class. Also by using [10, 4.5], we see that
G7 = hα, α1, α2, α3, α4|[αi, α] = αi+1, αp = α4, α1(p) = αi+1(p) = 1 (i = 1,2,3)i.
On Setting H = hα1, α2, α3, α4i, we deduce that H is abelian and |G7/H| = p, which completes
the proof by using Corollary 2.9(i).
(iii) By Table 1, G8 is of maximal class. Also by [10, 4.5], we have the following presentation forG8:
G8 =hα, α1, α2, α3, α4|[αi, α] =αi+1,[α1, α2] =α4, αp =α1(p) =αi+1(p) = 1 (i= 1,2,3)i. Moreover
γi(G) =hαi, . . . , α4i for 2≤i≤4 and soP1 =hα1, α2, α3, α4i since [α1, γ2(G)]≤γ4(G). This shows
that P1 is not abelian and the result follows from corollaries 2.7 and 2.9(ii).
References
[1] A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group,J. Algbera,298no. 2 (2006) 468-492. [2] A. Azad and Cheryl E. Praeger, Maximal subsets of pairwise non-commuting elements of three-dimensional general
linear groups,Bull. Aus. Math. Soc.,80no. 1 (2009) 91-104.
[3] A. Azad, S. Fouladi and R. Orfi, Maximal subsets of pairwise non-commuting elements of some finitep-groups,Bull. Iranian Math. Soc.,39no. 1 (2013) 187-192.
[4] Y. Berkovich,Groups of prime power order,1, Walter de Gruyter, Berlin, 2008.
[5] A. Y. M. Chin, On non-commuting sets in an extraspecialp-group,J. Group Theory,8no. 2 (2005) 189-194. [6] S. Fouladi and R. Orfi, Maximal subsets of pairwise non-commuting elements of somep-groups of maximal class,
Bull. Aust. Math. Soc.,84no. 3 (2011) 447-451.
[7] S. Fouladi and R. Orfi, Maximum size of subsets of pairwise non-commuting elements in finite metacyclicp-groups, Bull. Aust. Math. Soc.,87no. 1 (2013) 18-23.
[8] TheGAPGroup,GAP- -Groups, Algorithms, and Programming, Version 4.4.12; 2008. (http://www.gap-system.org). [9] C. R. Leedham-Green and S. McKay,The Structure of Groups of Prime Power Order,27, of London Mathematical
Society Monographs, New Series, Oxford University Press, Oxford, 2002.
[10] R. James, The Groups of Orderp6(pan odd prime),Math. Comp.,34no. 150 (1980) 613-637.
[12] L. Pyber, The number of pairwise non-commuting elements and the index of the centre in a finite group,J. London Math. Soc. (2),35(1987) 287-295.
[13] D. M. Rocke,p-groups with abelian centralizers,Proc. London Math. Soc. (3),30(1975) 55-75.
Reza Orfi
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8- 8349, Iran.
