IRJET - Triple Factorization of Non-Abelian Groups by Two Minimal Subgroups
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This document presents research on the triple factorization of non-abelian groups by two minimal subgroups. It begins with an abstract discussing the triple factorization of a group G as G = ABA, where A and B are proper subgroups of G. It then studies the triple factorizations of two classes of non-abelian finite groups: the dihedral groups D2n and the projective special linear groups PSL(2,2n). Several lemmas and a main theorem are presented regarding the triple factorizations and related rank-two coset geometries of these groups. In particular, the theorem discusses conditions for the existence of non-degenerate triple factorizations of D2n and properties of the associated rank-two coset
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 07 Issue: 02 | Feb 2020 www.irjet.net p-ISSN: 2395-0072
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TRIPLE FACTORIZATION OF NON-ABELIAN
GROUPS BY TWO MINIMAL SUBGROUPS
Sudha Lakshmi. , Parvatha Varthini. (Maters in Mathematics)
Thassim Beevi Abdul Kadar College for Women, Kilakarai, Ramanathapuram, Tamilnadu, India.
-----------------------------------------------------------------------------***-------------------------------------------------------------------------
Abstract.The triple factorization of a group G has been studied recently showing that G = ABAfor some proper subgroups A and B
of G, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was de
fined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups D2n and P SL (2,
2n) for their triple factorizations by finding certain suitable minimal subgroups, which these subgroups are define with original
generators of these groups. The related rank-two coset geometries motivateus to define the rank-two coset geometry graphs
which could be of intrinsic tool on the study of triple factorization of non-abelian groups.
Keywords: Rank-two geometry, triple factorization, dihedral groups, projective special linear groups.
I. Introduction
The factorization of a finite group G as the inner product G = ABA where, A and B are proper subgroups of G, the notation T
= (G, A, B) is used for a triple factorization of the group G. finite simple groups and their automorphism groups were
studied .The aim of this paper is to study the rank-two coset geometry by defining a graph, which is named a rank-two
coset geometry graph. The notation Γ(G, A, B) will be used for this graph, where G = ABA. Our computational results based on
the study of two classes of non-abelian groups D2n (the dihedral group of order 2n) and the projective special linear groups P
SL(2, 2n), (n ≥ 3). The nice and very interesting presentation of projective special linear groups may be found in ([5, 6, 7]) and
the related references.
It is necessary to recall that for studying the triple factorization of groups the important tools come from permutation group
theory and we recall some of them which will be useful in our proofs. The set of all permutations of a set Ω is the symmetric
group on Ω, denoted by Sym(Ω), and a subgroup of Sym(Ω) is called a permutation group on Ω. If a group G acts on Ω we
denote the induced permutation group of G by GΩ, a subgroup of Sym(Ω). We say that G is transitive on Ω if for all α, β ∈ Ω
there exists g ∈ G such that αg = β. For a transitive group G on the set Ω, a nonempty subset ∆ of Ω is called a block for G if for
each g ∈ G, either ∆g = ∆, or ∆g ∩ ∆ = ∅; in this case the set Σ = {∆g|g ∈ G} is said to be a block system for G. The
group G induces a transitive permutation group GΣ on Σ, and the set stabi- lizer G∆ induces a transitive permutation group G∆
on ∆. If the only blocks for G are the singleton subsets or the whole of Ω we say that G is primitive, and otherwise G is
imprimitive.
II. PRELIMINARIES
Definition 2.1. A triple factorization T = (G, A, B) of a finite group G is called degenerate if G = AB or G = BA. Otherwise, T = (G,
A, B) is called a non-degenerate triple factorization. A group with a triple factorization T = (G, A, B), is sometimes called an
ABA-group.
Definition 2.2. Let P and L be the sets of right cosets of the proper subgroups A and B of a finite group G, respectively. The
property ∗ between the elements of P and L which is named a ”non-empty inter- section relation” is defined as follows:
Ax ∗ By ⇐⇒ Ax ∩ By = ∅
Then (Ω = P ∪ L, ∗) is called a rank-two coset geometry and will be denoted by Cos(G, A, B).
In a rank-two coset geometry, if the property ∗ holds between two members Ax ∈ P and By ∈ L, then we say
that these members are incident, and in this case the pair (Ax, By) is called a flag of rank-two coset geometry.](https://image.slidesharecdn.com/irjet-v7i2420-201123063148/85/IRJET-Triple-Factorization-of-Non-Abelian-Groups-by-Two-Minimal-Subgroups-1-320.jpg)
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 07 Issue: 02 | Feb 2020 www.irjet.net p-ISSN: 2395-0072
© 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2012
thecompletebipartitegraph.BythesamemethodonemayseethatifT=(D2n,A,B)isadegeneratetriplefactorization
fortwodistinctsubgroupsAandB,thenΓ(D2n,A,B)=Kr,s where,randsaretheindicesofthesubgroupsAandB,re-
spectively.Fortheinversecase,letΓ(D2n,B,C)=Kp,q.the bydefinitionofrank-twogeometrygraphpandqaretheor-
dersoftwodistinctpropersubgroupsB=<ar,aib>and C=<as,ajb>,where|D2n:B|=r,|D2n:C|=sand
thesetofrightcosetsofBandCare{B,Ba,Ba2,...,Bar−1} and{C,Ca,Ca2,...,Cas−1},respectively.Nowbyconsidering
theelementsofsubgroupsB,CandD2nonemayseethat D2n=BCandthetriplefactorizationisdegenerate.✷
Lemma 3.2.
EverysubgroupofD2n(n≥3),iscyclicoradihedral groupsuchthat:
(i)thecyclicsubgroupsare<ad>,whered|nand|D2n:<ad>|=2d,
(ii)thedihedralsubgroupsare<ad,aib>,whered|n,and0≤i≤d−1,and|D2n:<ad,aib>|=d
(iii) let n be odd and m|2n. For odd values of m there are m sub- groups of index m inD2n However, if m is even there
is exactly one subgroup of index m,
(iv) let n be even and m|2n. For odd values of m there are m sub- groups of index m. If m is even and doesn’t divide n,
there is only one subgroup of index m. Finally, if m is even and m|n, there are exactly m + 1 subgroups of index m.
There are also certain obvious relations in D2n Indeed, for every integer i = 1, 2, ..., n, the following relations hold in D2n:
baib=a−i,aiba−i=a2ib,(aib)b(aib)−1=a2ib,aiba−i=b
Lemma 3.3.
Let A and B be two proper subgroups of a group G, and consider the right coset action of G on ΩA = {Ag|g ∈ G}.
Set α = A ∈ΩA. Then T = (G, A, B) is a triple factorization if and only if the B-orbit αBintersectsnontriviallyeachGα-orbitinΩA.
Lemma 3.4 Let A and B be two proper subgroups of a group G and consider the right coset action of G on ΩA = {Ag|g ∈
G}. Set α = A ∈ΩA Then, T = (G, A, B) is a triple factorization if and only if for all g ∈ G there exists elements b ∈ B, a ∈ A
such that Ab = Aga.
Lemma 3.5. For any two proper and distinct subgroups A and B of if T = (D2n, A, B) is a degenerate (non-
degenerate) triple factorization for D2n then T = (D2n, B, A) is also a degenerate (non- degenerate) triple factorization
for D2n. Moreover, D2n = ABA = BAB.
Proof. The proof is easy by using Lemma 2.3 and the relations of D2n
Lemma 3.6. There are exactly P2(n)presentations for the group
P SL(2,2
n
), (n ≥ 3), where P2(n)(n) = 1 d|n µ(n )2d and µ is the Mobius
Proof. In the relation xn=yxyan−1xyan−2...xya0of Sinkov’s presentation, every choice of a0,a1,...,an−1yields an irreducible
polynomial over GF (2) of degree n. On the other hand by the elementary results of [12], the number of such polynomials is
P2(n)(n) = 1 P d|n µ(n )2d, where µ is the Mobius function. where, for at least a primitive α of GF (2
n)), m(α) = 0. So, the
number of distinct presentations for P SL(2, 2n)) is P2(n).](https://image.slidesharecdn.com/irjet-v7i2420-201123063148/85/IRJET-Triple-Factorization-of-Non-Abelian-Groups-by-Two-Minimal-Subgroups-3-320.jpg)
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 07 Issue: 02 | Feb 2020 www.irjet.net p-ISSN: 2395-0072
© 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2013
Lemma 3.7. For every integer n ≥ 3, the last relation of the pre- sentation of P SL(2,2n) will be reduced to
xn=yxn−1yxyorxn= yxn−2yx2y, if n is even either is odd.
Proof. For n = 3, P2(3) = 2 (the number of irreducible polynomials of degree 3 over GF (2)) and one of these polynomials
is the trinomial m(x) =x3+x2+1. For n = 4, P2(4) = 3 and one of these polynomials is the trinomial m(x) = x4+x+1.. On the
other hand by using the results of [14] we deduce that, for every integer n ≥ 3, at least one of the irreducible polynomials of
degree n is a trinomial, and this trinomial is in the form m(x) = xn+x2+1or m(x) = xn+ x + 1 when n is odd either n is even,
respectively. Now, by considering the coefficients of this trinomials we see that the relation
xn=yxyan−1xyan−2...xya0for even values of n is equal to xn=yxn−1yxyand for the odd values of n is equal to xn=yxn−2yx2y.
Lemma 3.8. Let n ≥ 3. By considering the types of minimal subgroups of G = P SL(2, 2n
), if the subgroup H is of type
E2nnZ2nand the subgroup K is of type D2(2n+1)orD2(2n−1) then, there exist elementsh ∈ H, k ∈ K and g ∈ G such that
Hgh = Hk.
Proof. For every integer n ≥ 3, consider the minimal subgroups and E2nnZ2n−1andK=D2(2n+1). For every elements g ∈ G, h
∈ H and k ∈ K if Hgh = Hk, then Hghk−1=HIndeed, for every elements g, h and k from G, H and K, the element ghk−1
doesn’t belong to H, which is a contraction, because for three elements h, k and g0=hkh−1 from H, K and G,
g0hk−1=(hkh−1)hk−1= h ∈ H. So, there exist elements h ∈ H, k ∈ K and g ∈ G such that Hgh = Hk.
References
1. S. H. Alavi and C. E. Praeger, On Triple Factorisations of Finite Groups, J.
Group Theory, 14 (2011), 341-360.
2. B. Amberg, S. Franciosi, and F. De Giovanni, Products of groups, Oxford Uni- versity Press, 1992.
3. F. Buekenhout (editor), Handbook of Incidence Geometry, Building and Foun- dations, Elsevier, Amesterdam, 1995.
4. F. Buekenhout, J. De Saedeleer and D. Leemans, On the rank-two geometries of the groups P SL(2, q): Part II, Ars
Mathematica Contemporanea 6 (2013),
5. C. M. Campbell and E. F. Robertson, A deficiency zero presentation for SL(2, p), Bull. LondonMath. Soc., 12 (1980), 17-20.](https://image.slidesharecdn.com/irjet-v7i2420-201123063148/85/IRJET-Triple-Factorization-of-Non-Abelian-Groups-by-Two-Minimal-Subgroups-4-320.jpg)
IRJET - Triple Factorization of Non-Abelian Groups by Two Minimal Subgroups
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 02 | Feb 2020 www.irjet.net p-ISSN: 2395-0072 © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2010 TRIPLE FACTORIZATION OF NON-ABELIAN GROUPS BY TWO MINIMAL SUBGROUPS Sudha Lakshmi. , Parvatha Varthini. (Maters in Mathematics) Thassim Beevi Abdul Kadar College for Women, Kilakarai, Ramanathapuram, Tamilnadu, India. -----------------------------------------------------------------------------***------------------------------------------------------------------------- Abstract.The triple factorization of a group G has been studied recently showing that G = ABAfor some proper subgroups A and B of G, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was de fined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups D2n and P SL (2, 2n) for their triple factorizations by finding certain suitable minimal subgroups, which these subgroups are define with original generators of these groups. The related rank-two coset geometries motivateus to define the rank-two coset geometry graphs which could be of intrinsic tool on the study of triple factorization of non-abelian groups. Keywords: Rank-two geometry, triple factorization, dihedral groups, projective special linear groups. I. Introduction The factorization of a finite group G as the inner product G = ABA where, A and B are proper subgroups of G, the notation T = (G, A, B) is used for a triple factorization of the group G. finite simple groups and their automorphism groups were studied .The aim of this paper is to study the rank-two coset geometry by defining a graph, which is named a rank-two coset geometry graph. The notation Γ(G, A, B) will be used for this graph, where G = ABA. Our computational results based on the study of two classes of non-abelian groups D2n (the dihedral group of order 2n) and the projective special linear groups P SL(2, 2n), (n ≥ 3). The nice and very interesting presentation of projective special linear groups may be found in ([5, 6, 7]) and the related references. It is necessary to recall that for studying the triple factorization of groups the important tools come from permutation group theory and we recall some of them which will be useful in our proofs. The set of all permutations of a set Ω is the symmetric group on Ω, denoted by Sym(Ω), and a subgroup of Sym(Ω) is called a permutation group on Ω. If a group G acts on Ω we denote the induced permutation group of G by GΩ, a subgroup of Sym(Ω). We say that G is transitive on Ω if for all α, β ∈ Ω there exists g ∈ G such that αg = β. For a transitive group G on the set Ω, a nonempty subset ∆ of Ω is called a block for G if for each g ∈ G, either ∆g = ∆, or ∆g ∩ ∆ = ∅; in this case the set Σ = {∆g|g ∈ G} is said to be a block system for G. The group G induces a transitive permutation group GΣ on Σ, and the set stabi- lizer G∆ induces a transitive permutation group G∆ on ∆. If the only blocks for G are the singleton subsets or the whole of Ω we say that G is primitive, and otherwise G is imprimitive. II. PRELIMINARIES Definition 2.1. A triple factorization T = (G, A, B) of a finite group G is called degenerate if G = AB or G = BA. Otherwise, T = (G, A, B) is called a non-degenerate triple factorization. A group with a triple factorization T = (G, A, B), is sometimes called an ABA-group. Definition 2.2. Let P and L be the sets of right cosets of the proper subgroups A and B of a finite group G, respectively. The property ∗ between the elements of P and L which is named a ”non-empty inter- section relation” is defined as follows: Ax ∗ By ⇐⇒ Ax ∩ By = ∅ Then (Ω = P ∪ L, ∗) is called a rank-two coset geometry and will be denoted by Cos(G, A, B). In a rank-two coset geometry, if the property ∗ holds between two members Ax ∈ P and By ∈ L, then we say that these members are incident, and in this case the pair (Ax, By) is called a flag of rank-two coset geometry.
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 02 | Feb 2020 www.irjet.net p-ISSN: 2395-0072 © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2011 Definition 2.3. The rank-two coset geometry graph of a finite non- abelian group G will be denoted by Γ(G, A, B), is an undirected graph with the vertex set P ∪ L and two points Ax and By are adjacent if and only if Ax ∩ By = ∅ where, G = ABA. III.MAIN RESULT Theorem 3.1. Let G = = ha, b| = = (ab)2 = 1i be the dihedral group of order 2n. Then, (1) For n = 3k, (k = 1, 2, ...), there are at least two proper dihedral subgroups B and C of G such that G=BCB (non-degenerate triple factorization). (2) For n = 2k,, (k = 1, 2, ...), there is no non-degenerate triple factorization for G. (3) For the prime values of n ≥ 5, there is no non-degenerate triple factorization for G. (4) The graph associated to a triple factorization T = (G, A, B) of G, (Γ(G, A, B)) is bipartite graph if and only if the factorization is degenerate. Proof: (1)Forn=3k,(k=1,2,3,...),D2n=ha,b|a3k=b2=(ab) 2 =1i anditsdihedralsubgroupsareintheform<ad,aib>where, d≥3,d|n=3kand0≤i≤d−1.NowifB=<ar,aib> andC=<as,ajb>betwodistinctdihedralsubgroupofD2n=D2(3k) such that |B||C||B| ≥ 2n, thenforsomei,j,l,m,nthat,0≤i,j,l≤n−1and0≤m,n≤1,thereexistelementsx=aibm∈B,y=ajbn∈Candg=al∈D2nsuchthat By=Bgx.So,byLetT=(D2n,A,B)isatriple factorizationofD2n,andbyusingtherelationsbaib=a−i, aiba−i=a2ib,(aib)b(aib)−1=a2ibandaiba−i=b,(0≤i≤ n−1)ofD2nwegetthatforevery0≤r,s,l≤n−1and0≤α,β,γ≤1,thewordarbαasbβalbγofBCBisoneofthe elementsofD2n.So,thistriplefactorizationisnon-degenerate andD2n=BCB=CBC.(2) Forn=2k,(k=1,2,3,...),byLemma2.1,thenumberofnon- trivialcyclicanddihedralsubgroupsofD2niskand2k+1−2,respectively.Inthecasek=1,thenon-trivialcyclicsubgroupof D4isA=<a>={1,a}andthenontrivialdihedralsubgroups areB=<a2,a0b>=<1,b>={1,b}andC=<a2,a1b>=< 1,ab>={1,ab},suchthatbyusingtherelationsofD2nwe get,AB=BA=AC=CA=BC=CB=D4.And foreveryk≥2,itiseasytoseethatforthecyclicsubgroup A=<a1>andforanytwodistinctnontrivialdihedralsub- groupsBandCsatisfyingB*C,C*Band|B||C||B|≥2n wegetAB=BA=AC=CA=BC=CB=D2n.Hence,the triples(D2n,A,B),(D2n,A,C)and(D2n,B,C)aredegenerate triplefactorizations. (3)Fortheprimevaluesofn≥5,thenumberofnontrivialcyclic anddihedralsubgroupsofD2nare1andn,respectively,where A=<a>istheonlynontrivialcyclicsubgroupandforevery i(i=0,1,...,n−1),Bi=<an,aib>isanontrivialdihedral subgroup.ByusingtherelationsofD2nonemayseethatfor every1≤i,j≤n−1,ABiA=ABi=D2nbutBiBjBi=D2n. Thus,inthiscasethereisnonon-degeneratetriplefactorization forD2n. (4)By(2)and(3),T=(D2n,A,Bi)isadegeneratetriplefactor- izationofD2nwhere,A=<a>istheonlycyclicsubgroup ofD2nofindex2andBi=<an,aib>,(i=0,1,...,n−1)is adihedralsubgroupofindexn,wheren≥5isaprimeand thesetofdistinctrightcosetsofAandBiare{A,Ab}and{Bi,Bia,Bia2,...,Bian−1},respectively.Byusingtherelations ofD2nwegetthatforevery0≤i,k≤n−1,A∩Biakand Ab∩Biakarenotempty.Sobythedefinitionofrank-twocoset geometry,foreveryi,(i=0,1,...,n−1),eachcosetofAis adjacenttoallcosetsofBi.Therefore,Γ(D2n,A,Bi)=K2,n−1,
- 3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 02 | Feb 2020 www.irjet.net p-ISSN: 2395-0072 © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2012 thecompletebipartitegraph.BythesamemethodonemayseethatifT=(D2n,A,B)isadegeneratetriplefactorization fortwodistinctsubgroupsAandB,thenΓ(D2n,A,B)=Kr,s where,randsaretheindicesofthesubgroupsAandB,re- spectively.Fortheinversecase,letΓ(D2n,B,C)=Kp,q.the bydefinitionofrank-twogeometrygraphpandqaretheor- dersoftwodistinctpropersubgroupsB=<ar,aib>and C=<as,ajb>,where|D2n:B|=r,|D2n:C|=sand thesetofrightcosetsofBandCare{B,Ba,Ba2,...,Bar−1} and{C,Ca,Ca2,...,Cas−1},respectively.Nowbyconsidering theelementsofsubgroupsB,CandD2nonemayseethat D2n=BCandthetriplefactorizationisdegenerate.✷ Lemma 3.2. EverysubgroupofD2n(n≥3),iscyclicoradihedral groupsuchthat: (i)thecyclicsubgroupsare<ad>,whered|nand|D2n:<ad>|=2d, (ii)thedihedralsubgroupsare<ad,aib>,whered|n,and0≤i≤d−1,and|D2n:<ad,aib>|=d (iii) let n be odd and m|2n. For odd values of m there are m sub- groups of index m inD2n However, if m is even there is exactly one subgroup of index m, (iv) let n be even and m|2n. For odd values of m there are m sub- groups of index m. If m is even and doesn’t divide n, there is only one subgroup of index m. Finally, if m is even and m|n, there are exactly m + 1 subgroups of index m. There are also certain obvious relations in D2n Indeed, for every integer i = 1, 2, ..., n, the following relations hold in D2n: baib=a−i,aiba−i=a2ib,(aib)b(aib)−1=a2ib,aiba−i=b Lemma 3.3. Let A and B be two proper subgroups of a group G, and consider the right coset action of G on ΩA = {Ag|g ∈ G}. Set α = A ∈ΩA. Then T = (G, A, B) is a triple factorization if and only if the B-orbit αBintersectsnontriviallyeachGα-orbitinΩA. Lemma 3.4 Let A and B be two proper subgroups of a group G and consider the right coset action of G on ΩA = {Ag|g ∈ G}. Set α = A ∈ΩA Then, T = (G, A, B) is a triple factorization if and only if for all g ∈ G there exists elements b ∈ B, a ∈ A such that Ab = Aga. Lemma 3.5. For any two proper and distinct subgroups A and B of if T = (D2n, A, B) is a degenerate (non- degenerate) triple factorization for D2n then T = (D2n, B, A) is also a degenerate (non- degenerate) triple factorization for D2n. Moreover, D2n = ABA = BAB. Proof. The proof is easy by using Lemma 2.3 and the relations of D2n Lemma 3.6. There are exactly P2(n)presentations for the group P SL(2,2 n ), (n ≥ 3), where P2(n)(n) = 1 d|n µ(n )2d and µ is the Mobius Proof. In the relation xn=yxyan−1xyan−2...xya0of Sinkov’s presentation, every choice of a0,a1,...,an−1yields an irreducible polynomial over GF (2) of degree n. On the other hand by the elementary results of [12], the number of such polynomials is P2(n)(n) = 1 P d|n µ(n )2d, where µ is the Mobius function. where, for at least a primitive α of GF (2 n)), m(α) = 0. So, the number of distinct presentations for P SL(2, 2n)) is P2(n).
- 4. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 07 Issue: 02 | Feb 2020 www.irjet.net p-ISSN: 2395-0072 © 2020, IRJET | Impact Factor value: 7.34 | ISO 9001:2008 Certified Journal | Page 2013 Lemma 3.7. For every integer n ≥ 3, the last relation of the pre- sentation of P SL(2,2n) will be reduced to xn=yxn−1yxyorxn= yxn−2yx2y, if n is even either is odd. Proof. For n = 3, P2(3) = 2 (the number of irreducible polynomials of degree 3 over GF (2)) and one of these polynomials is the trinomial m(x) =x3+x2+1. For n = 4, P2(4) = 3 and one of these polynomials is the trinomial m(x) = x4+x+1.. On the other hand by using the results of [14] we deduce that, for every integer n ≥ 3, at least one of the irreducible polynomials of degree n is a trinomial, and this trinomial is in the form m(x) = xn+x2+1or m(x) = xn+ x + 1 when n is odd either n is even, respectively. Now, by considering the coefficients of this trinomials we see that the relation xn=yxyan−1xyan−2...xya0for even values of n is equal to xn=yxn−1yxyand for the odd values of n is equal to xn=yxn−2yx2y. Lemma 3.8. Let n ≥ 3. By considering the types of minimal subgroups of G = P SL(2, 2n ), if the subgroup H is of type E2nnZ2nand the subgroup K is of type D2(2n+1)orD2(2n−1) then, there exist elementsh ∈ H, k ∈ K and g ∈ G such that Hgh = Hk. Proof. For every integer n ≥ 3, consider the minimal subgroups and E2nnZ2n−1andK=D2(2n+1). For every elements g ∈ G, h ∈ H and k ∈ K if Hgh = Hk, then Hghk−1=HIndeed, for every elements g, h and k from G, H and K, the element ghk−1 doesn’t belong to H, which is a contraction, because for three elements h, k and g0=hkh−1 from H, K and G, g0hk−1=(hkh−1)hk−1= h ∈ H. So, there exist elements h ∈ H, k ∈ K and g ∈ G such that Hgh = Hk. References 1. S. H. Alavi and C. E. Praeger, On Triple Factorisations of Finite Groups, J. Group Theory, 14 (2011), 341-360. 2. B. Amberg, S. Franciosi, and F. De Giovanni, Products of groups, Oxford Uni- versity Press, 1992. 3. F. Buekenhout (editor), Handbook of Incidence Geometry, Building and Foun- dations, Elsevier, Amesterdam, 1995. 4. F. Buekenhout, J. De Saedeleer and D. Leemans, On the rank-two geometries of the groups P SL(2, q): Part II, Ars Mathematica Contemporanea 6 (2013), 5. C. M. Campbell and E. F. Robertson, A deficiency zero presentation for SL(2, p), Bull. LondonMath. Soc., 12 (1980), 17-20.