SYMMETRIC GENERATING SET OF THE GROUPS A,./, AND $2,./

Internat. J. Math. & Math. Sci.

VOL. 21 NO. (1998) 139-144

139

SYMMETRIC GENERATING SET OF THE GROUPS A,./, ^AND $2,./

USING S. ^{AND AN ELEMENT OF ORDER TWO}

A.M. HAMMAS

Department

ofMathematics

College ofEducation

P.O.Box

344 King Abdulaziz University

Madinah, Saudi Arabia

(Received March 18, 1996 and in revised form August 28, 1996)

ABSTRACT. In

thispaperwewillshowhowto generate ingeneral

A2n +

^and

S2n

+1usingacopy of S_n and anelement of order 2 in

A2n+l

^or

S2n+l

^{for all}positive integers^n_>2.

We

willalso show howto generate

A2n+l

^and

S2n+l

symmetrically usingnelements eachoforder 2.

KEY WORD AND

PHRASES: Symmetric generators, Involution, Double transitivegroups,

Group

presentation

1991AMS SUBJECT

CLASSIFICATION CODES:

20F05 1.

INTRODUCTION

Itisshownby

Hammas 1]

that

A2n

^{+ canbe}presented as

G=A2n+I ^{<X, Y,} TI "<X,

Y>=S_n,

T2= [T, Sn_I]

^{=I >}

for n=4, 6, where

IT, Sn_l]

^meansthat

^Tcommutes

^withYand

withX-2yx, (the

generatorsof

Sn_l).

The relations of the symmetricgroup

S

_n <

X,Y

>of degreenarefoundin

Coxeter

and

Moser[2]. Some

relations mustbe addedtothe presentation of

A2n

^{+ in ordertocompletethecoset}enumeration. Also,it has been shown by

Hammas

that forn 4,6, thegroup

A2n

+ ^canbesymmetricallygenerated byn elements

TO, T1, Tn_

1, each oforder 2,oftheform

Ti= ^T Xi _X -z’TX ^i,

^where

^T

^and

^X

^satisfythe

relations ofthe group

A2n+l.

^Theset

^{T

0,

T Tn_

^iscalled a symmetricgeneratingsetof

A2n+l

(seesection

3).

In

this paper, we give a generalization ofthe results obtained by

Hammas [1]

^forall n>2.

Moreover

a

proof

isgiventoshowthatthe

group

0

<X,Y, TI .<X,Y>=S n,T

2

^=[T, Sn_I]=I>

is either

A2n+l

ifn is even or

$2n+1

^{ifnisoddfor all n>2.} We give permutationsthatgenerate

A2n+l

and

S2rt+

^{for alln}>_ 2 which satisfytheconditionsgivenin thepresentation ofthegroup

G.

Further,we prove that

G

canbe symmetricallygenerated bynpermutations,each of order 2,satisfyingthe condition given in remark 2.4.

Our

research is motivatedbythe aimof showing groupsin theirmost"natural" roleactingon(or permuting) the members of asymmetric generatingset. Theauthorhasappliedthemethodtoobtainthe symmetricgeneratingsetsandthe presentationsofthefollowingfinitesimple

groups:

Tits group

2F4(2)’, Janco

groups

J!

^and

^J2,

^{Mathieu groups}

Ml2and ^M24,

^{and someofthelinear}

groups PSL(2,q).

For

moredetails,see

Hammas ].

2.

PRELIMINARY RESULTS

In

thissection, we give someofthepreliminaryresultstobe used in later sections. Theproofs of these results can befound inmany references,seeforexample

[2], [4],

and

[5].

LEMMA

2.1.

Let

_< a b_< n be integerswherenisodd.

Let G

be thegroup generated by the n-cycle

(1,

2,

...,n)

andthe3-cycle n,a,

b).

Ifthehighestcommonfactor

hef(

n,a, b 1, then

G =A

_n.

LEMMA

2.2.

Let

nbe anodd integer andlet

G

bethe group generated bythen-cycle 1, 2, n and thek-cycle 1,2 k

).

If <k<nand kisanodd integer,then

G A n-

PROOF. Let

r

=(1,2,3

,n),and

=(1,2 ,k).

Sincethecommutator

[6, z]=(1,2,k+l),

^thenby

Lemma

2.1,G_=A

LEMMA

2.3.

Let G

be thegroup generated by n-cycle

(1,2 n)

and theinvolution

(n,1)(id’)

^{for <i,} j <n.Ifn >9isanoddinteger then

G A n.

REMARK

2.4. The main condition used in

Hammas ],

^{which wearegoingto}use in thispaper,isthat

T

commuteswiththegenerators ofthegroup

Sn.

1"

3.

SYMMETRIC GENERATING SETS

Let G

be agroupandlet

F T

_0,

T Tn-1

^{beasubsetofG, where}

^{T T} Xi X -iTX

for all 0, n--1.

Let S

_nbethe normalizerof theset

F

in

G,

which is acopy of the symmetric

group

of degree ^n.

We

define

F

tobeasynunetric generating^setof

G

ifandonlyif

G

<

F

> and

S

_n permutes

F

doubly transitively by conjugation. Equivalently,

F

is realizable as an innerautomorphism.

4.

PERMUTATIONAL GENERATING SET OF A2n+l

^and

S2n+l

THEOREM

4.1.

A2n+l ^(S2n+l)

^{can be}

^generated

^usinga

^copy

^of

^S

n andanelement oforder 2in

A2n+l ^(S2n+l)

ifn is even

(odd)

for alln>2.

PROOF. Let X=(1,

2

n)(n+l,

n+2 2n),

Y=(n-1, n)(2n-1, 2n)

and

T (1, 2n+1)(2, n+2) (n, 2n)

be threepermutations;the first is ofordern, thesecondand the third areof order 2.

Let H

bethe

group

SYMMETRIC GENERATING SET OF GROUPS 141

generated by

X

andY.

By

the Burnside and

Moore

Theorem(see CoxeterandMoser

[2]

),thegroup

H

is the symmetric group S_n Let

(3

_{be the}group generated by X,

Y

and T. Consider thecommutator

rl

=[

^X,

T ],

^whichhasthe form _rl

,n+

1,2n+ ,n+2,2). Then

3

X

rl rl 1,2n+ )(2,n+3,3)(n+ ,n+2) ^a.

Therefore a2

(2,3,n+3).

Hence

2)X

^-1

Xrl

^{(or (1,2}

^n,n+

2n,2n+ 1).

Let

13

^Xrl(o2X

^{Let K}

^<

^13,

^{c2,T >be asubgroupof}

^(3.

Since the highest common factor hef(2,3,n+3) ^1, then byLemma2.1

<13,et

2

>=A2n+l.

^{Nowifnis aneveninteger,}

thenK=A2n+l"

Since

X,

YandTareevenpermutationsthen

K =Q A2n+l.

^{Also,ifn isanoddinteger,then}

^T

^isanodd

permutationandtherefore

K (3 S2n

+1"

5.

SYMMETRIC GENERATING

SET

OF A2n+l

^and

S2n+l

THEOREM

5.1.Let X,Yand

T

be thepermutationsdescribed in Theorem 4.1.

Let F ={T0,T Tn_ },

where

T T X

and 0,1 n-1.Ifn is an eveninteger,then theset

F

generates thealternatinggroup

A2n+l

symmetrically, while if n is an odd integer, then theset

F

generates the symmetricgroup

S2n

+ symmetrically.

PROOF. Let To=(1,2n+l)(2,n+2)...

^(n,2n),

Tl=(1,n+l)(2,2n+l)... (n,2n), Tn_ ^T Xn-1

=(

n,2n+

(1,n+l)

(n-l,2n-1).

Let

H <F

>.

We

claimthat if n is an eveninteger,then

H=A2n+I

and if n is an odd integer, then

H=S2n+I. ^To

^{show this, suppose firstthat n is aneveninteger.}Consider the element

n-1

a=H ^{T Xi}

i=0

It isnotdifficulttoshow that c

=(1,2,n+2,n+3,3,4,n+4,n+5,5,6

2n,2n+ 1,n+

1)

and it is acycleoflength 2n+l. Let

[3=ToT

1. It is clear that

fS=(1,2,n+2,2n+2,n+l).

Let

HI= <ct,[3>. We

claim that

H1 A2n+

1" To provethis, let0 be themappingwhichtakes the element in theposition of thecycle

ct into the element ofthecycle (1,2

2n+l).

Underthismapping,thegroup

H

will bemappedinto the group 0

(H 1)=

^{<(1,2 2n+}

1),(1,2,3,2n,2n+l)>

which is, by

Lemma

2.2, the alternating group

A2n+l.

^Hence

^H=H O(H1)=A2n+I.

Second, supposethatnisanoddinteger. Considerthe element

n

=H ^{T Xi}

i=l

It is not difficult toshow that 5 =(1,2n+l,2,n+3.4,n+5.6,n+7 2n and it is acycleoflengthn+l. Let

T T

t 6

T

_O Since 6 (2,2n+1,3.n+4,5,n+6 n,n+l),then

=(^1,2n+ ,n+3,3,4,n+4,n+5

n,n+

,n+2,2)

which is a cycle of length2n+l.

Let [3 T T

₂ then

13

^=(2,n+2,3

)(4,2n+l)(n+3,n+4).

Therefore

13

²

=(2,3,n+2). Let H

₂

---< ^,13 2,T

₀>. Weclaimthat

H

₂

=_S2n+l. ^To

^provethis, let0 be themapping which takes the element in theposition of thecycle t ^intotheelement ofthecycle(1,2 2n+l).

Under this mapping thegroup

H

_{2 will be}mapped^intothegroup

0

(H2)=

^{<(1,2 2n+}

^1),(2n+ 1,4,2n),(1,2)(3,4)...(2n-3,2n-2)(2n,2n+

1) >.

Since thehef(2n+l,4,2n)=l,thenthegroup<(1,2 2n+1),(2n+l,4,2n)^>isthealternating group

A2n+l.

Since n is an oddinteger,then thepermutation

(1,2)(3,4)...(2n-3,2n-2)(2n,2n+l)

isanoddpermutation.

Therefore thegroup 0

(H2)is

the symmetricgroup

S2n+l. ^Hence

^H_=

^H

2^_=0

(H2)=-S2n

+ 1"

The set F described above satisfies the conditions of the group

G

givenin section 1. Itis important to notethat

F

musthaveexactly nelementseach of order2 to generate

A2n

+ ^or

S2n

⁺1"The following Theorem characterizes all groups obtained by removing m elementsofthe set

F

for some integer ^m.

THEOREM

^5.2. Let

T

and

X

be thepermutationsdescribed above and let

F ={ T1.T

2

T n}.

^Then,

removing m elements of the set

F

for all <m_<n-3,theresultingsetgenerates

S2(n__m)+l,

^removing

m=(n-2)

elements ofthe set F, the resulting set generatesthe dihedralgroup oforder 10

(D10),

^and

removingm=(n-1)elements oftheset

F.

the resultingsetgenerates the cyclicgroup C_2.

PROOF.

Usinginduction onn-m, ifn-re=l,then

F I={T1 ^}.

^Since

Tl=(1,rt+l)(2,2n+l)(3,n+3)...(n, ^2n),

then

F

generates

C

2. Ifn-m=2,then

F

₂

T1,T2}-

^Since

T

is thepermutationdescribedabove,

T2= (1,n+l)(2,n+2)(3,2n+l)...(n,2n),

and

T 1T2=(2,3,n+3,2n+ ^1,n+2),

then it is clear that

F

₂ generates

(r2 T3...Vk_

D10. ^Now

^supposethat

^<m

^<n-3.Ifn-m

^k.

^then

^{F k={T1 Tk}.}

^Assuming

^{a=T T}

^k,

thenforkanevenintegerwehave

ct=(2,3 ,n+4,5 ,n+6,7,n+8 k-

,n+

k,k+

,n+

k+1,2n+ ,k,n+k- ,k-2,n+k-3 4.n+3

,n+2)

which is a

permutation

of

length

2k+1" whileifkis an oddinteger, then

a

=(2,n+2,3,n+3,4,n+5,6,n+7,8,n+9

k- ,n+k,k+ ,n+k+ 1,2n+ ,k,n+k- ,k-2,n+k-3 5,

n+4,3),

SYMMETRIC GENERATING SET OF GROUPS 143

it isalsoapermutationoflength2k+l.Let 3=

T T1T2T

3 Since

13=(2,

n+3)(3,

n+2)(

4, n+4,2n+l), then

13 3=(2,n+3)(3,n+2). By Lemma

2.3, cand

13

^{3 generate}

A2k+l.

^Hencethegroup generated byc,

13

^{3 and}

^T

^isthe Symmetricgroup

S2k

+ 1- Therefore the Theorem is^truefor allm.

REMARK.

The aboveresults are summarized in thefollowingtable

n

G=<X, Y,T> <X, T>

<

F>

even

A2n+l A2n+l A2n+l

2 odd

S2n+l S2n+l S2n+l

where

A2n+l ^{<X,Y, TI} <X,Y> S

_n,

T

²

[T,Y] =[T,X-2yx] (XT)

^2n+l

(YTn_2)10>.

S2n

+

<X,Y, T <X,Y

>

S

_n,

T

²

=[T,Y =[T, x’ZYx] (XT) n(n+a)

(Y Tn_2) 10>.

From

the above, we can see that the orderofthe element

XT

isn(n+

1)

whennisanoddinteger.

As n gets larger, the order ofXTbecomesverylarge.

For

thisreason,Hammas

[1]

had beenunableto proceedforlargeodd valuesofn.

REFERENCES

[1] HAMMAS,

A. M.. "Symmetric Presentations

of

^SomeFinite

Group’s."

Ph.D.Thesis,Unive:sity of Birmingham,

May

1991.

[2] COXETER, H.S.M.

and

MOSER, W.O.J., "Generators

and relations

for

^Discrete

Groups,"

third ed.,Springer-Verlag, New York, 1972.

[3] AI-AMRI, IBRAHIM R

and

HAMMAS,

A.M.,"SymmetricGenerating

Set of

^the

^Groups Akn

+ 1 and

Skn

⁺

1,"

^{To appearin}Journal of King Abdulaziz University, Sciences.

[4] AI-AMRI, IBRAHIM R.,

"ComputationalMethodsinPermutation

Group

Theory,"Ph.D.Thesis, University ofSt.Andrews,September1992.

[5] HAMMAS,

^A. M. and

AI-AMRI, IBRAHIM R.,

"Symmetric GeneratingSet

of

^theAlternating

Groups A2n

+

1,"

^{Journalof KingAbdulaziz}University,Educ. Sci., Vol.

7(1994),

3-7.