www. heo yo g oups.i
In e na ional Jou nal o G oup Theo y
ISSN (p in ): 2251-7650, ISSN (on-line): 2251-7669
Vol. 9 No. 1 (2020), pp. 31-42.
c
2020 Uni e si y o Is ahan
www.ui.ac.i
THE NUMBER OF MAXIMAL SUBGROUPS AND PROBABILISTIC
GENERATION OF FINITE GROUPS
ADOLFO BALLESTER-BOLINCHES, RAM´
ON ESTEBAN-ROMERO∗, PAZ JIM´
ENEZ-SERAL AND
HANGYANG MENG
Communica ed by Pa izia Longoba di
Abs ac . In his su ey we p esen some signi ican bounds o he numbe o maximal subg oups
o a gi en index o a ini e g oup. As a consequence, new bounds o he numbe o andom gene a o s
needed o gene a e a ini e d-gene a ed g oup wi h high p obabili y which a e signi ican ly igh e
han he ones ob ained in he pape o Jaikin-Zapi ain and Pybe (Random gene a ion o ini e and
p o ini e g oups and g oup enume a ion, Ann. Ma h.,183 (2011) 769–814) a e ob ained. The esul s
o Jaikin-Zapi ain and Pybe , as well as o he esul s o Lubo zky, De omi, and Lucchini, appea as
pa icula cases o ou heo ems.
1. In oduc ion
All g oups in his su ey will be ini e. Le Gbe a d-gene a ed g oup. The mo i a ing ques ion o
his pape is: How many elemen s one should expec o choose uni o mly and andomly o gene a e G?
Le us call his numbe ε(G). The i s e e ence we ha e o his p oblem is due o Ne o [18, page 90],
who conjec u ed ha he p obabili y ha a andomly chosen pai o elemen s o Al (n) gene a es
Al (n) ends o 1 as n ends o in ini y, and ha he p obabili y ha a andomly chosen pai o
elemen s o Sym(n) gene a es Sym(n) ends o 3/4 as n ends o in ini y. Dixon p o ed in [10] ha
his conjec u e is ue, by showing ha he p opo ion o gene a ing pai s o Al (n) o Sym(n) is
MSC(2010): P ima y: 20P05; Seconda y: 20E07; 20E28
Keywo ds: Fini e g oup, maximal subg oup, p obabilis ic gene a ion, p imi i e g oup.
Recei ed: 10 Decenbe 2018y, Accep ed: 24 Janua y 2019.
∗Co esponding au ho .
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32 In . J. G oup Theo y, 9 no. 1 (2020) 31-42 A. Balles e -Bolinches, R. Es eban-Rome o, P. Jim´enez-Se al and H. Meng
g ea e han 1 −2/(ln ln n)2 o sufficien ly la ge n, whe e ln deno es he na u al loga i hm. He
also conjec u ed ha he p opo ion o pai s (x, y) o elemen s o a simple g oup G ha gene a e G
also ends o in ini y as |G| ends o in ini y. Kan o and Lubo zky [14] and Liebeck and Shale [15]
p o ed he alidi y o Dixon’s conjec u e: i Gis an almos simple g oup wi h socle S, he p obabili y
ha a pai o elemen s o Ggene a es a subg oup con aining S ends o 1 as |G| ends o in ini y.
Pome ance [20] analysed he expec ed numbe ε(G) o andom gene a o s o an abelian g oup G
and p o ed ha ε(G)≤d(G)+σ, whe e d(G) deno es he minimum ca dinal o a gene a ing se o G
and σ= 2.118456563 ··· is ob ained om he Riemann ze a unc ion. Fo abelian g oups, his bound
is op imal. Pak [19], mo i a ed by po en ial applica ions o he p oduc eplacemen algo i hm,
use ul o ob ain andom elemen s in a ini ely gene a ed g oup, s udied a ela ed in a ian .
De ini ion 1.1. Gi en a g oup G,ν(G)deno es he leas posi i e in ege ksuch ha he p obabili y
o gene a ing Gwi h k andom elemen s is a leas 1/e.
Pak p o ed in [19, Theo em 2.5] ha
1
eε(G)≤ν(G)≤e
e−1ε(G)
and conjec u ed ha ν(G) = O(d(G) log log|G|). He e he symbol log is used o deno e he loga i hm
o he base 2, wi h he con en ion ha log 0 = −∞.
Le {x1, x2, . . . , xk} ⊆ G. Then we ha e ha ⟨x1, x2, . . . , xk⟩ =Gi , and only i , he e exis s a
maximal subg oup Mo Gsuch ha ⟨x1, x2, . . . , xk⟩ ≤ M. This shows he ele ance o maximal
subg oups in he scope o p obabilis ic gene a ion o ini e g oups. Mo eo e , i he elemen s x1,
x2, . . .,xka e chosen independen ly and a andom wi h a uni o m dis ibu ion and Mis a maximal
subg oup o G, hen
P ob(⟨x1, x2, . . . , xk⟩ ≤ M) =
k
∏
i=1
P ob(xi∈M) = (|M|
|G|)k
=1
|G:M|k.
The e o e he numbe mn(G) o maximal subg oups o Go a gi en index nis also ele an in his
con ex . Lubo zky [16], by conside ing he numbe o chie ac o s in a gi en chie se ies, and De omi
and Lucchini [8], wi h he help o he numbe λ(G) o non-F a ini chie ac o s in a gi en chie se ies
o Gand by means o hei associa ed c owns, p o ed independen ly ha Pak’s conjec u e is alid.
Theo em 1.2 (Lubo zky, [16, Co olla ies 2.6 and 2.8]).I Gis a g oup wi h chie ac o s in a
gi en chie se ies, hen
mn(G)≤ ( +nd(G))n2≤ 2nd(G)+2.
Fu he mo e,
ν(G)≤1 + log log|G|
log i(G)+ max (d(G),log log|G|
log i(G))+ 2.02,
whe e i(G)deno es he smalles index o a p ope subg oup o G.
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In . J. G oup Theo y, 9 no. 1 (2020) 31-42 A. Balles e -Bolinches, R. Es eban-Rome o, P. Jim´enez-Se al and H. Meng 33
The p oo o Theo em 1.2 depends on some esul s o Mann and Shale [17] abou he numbe
o maximal subg oups o a gi en index o a g oup Gand on some s uc u al esul s abou p imi i e
g oups.
Theo em 1.3 (De omi and Lucchini [8, Theo em 20]).The e exis s a cons an csuch ha , o any
g oup G,ν(G)≤ ⌊d(G) + clog λ(G)⌋i λ(G)>1, o he wise, ν(G)≤ ⌊d(G) + c⌋, whe e λ(G)deno es
he numbe o non-F a ini chie ac o s in a gi en chie se ies o G.
He e he symbol ⌊x⌋is used o deno e he de ec in ege pa o x, ha is, he la ges in ege
numbe nwi h n≤x. The p oo o his esul depends on some esul s o Dalla Vol a and Lucchini
[6] and Dalla Vol a, Lucchini, and Mo ini [7] abou he numbe o gene a o s o powe s o ce ain
diagonal- ype subg oups o di ec p oduc s o copies o a monoli hic g oup.
The in a ian ν(G) is ela ed o he ollowing in a ian .
De ini ion 1.4. Gi en a g oup Gle
M(G) = max
n≥2lognmn(G) = max
n≥2
log mn(G)
log n.
He e lognxdeno es he loa i hm o he base no x, ha is, lognx= log x/ log n= ln x/ ln n.
Theo em 1.2 depends on he ollowing in e es ing esul .
Theo em 1.5 (Lubo zky, [16, P oposi ion 1.2]).
M(G)−3.5≤ν(G)≤ M(G)+2.02.
Jaikin-Zapi ain and Pybe ha e ob ained in [12] a ema kable esul ha gi es su p isingly explici
lowe and uppe bounds o ν(G) wi h in e es ing applica ions (see [5, Theo em 3]). Thei bounds
depend on he ollowing in a ian s.
De ini ion 1.6. Le Gbe a g oup. The symbol l(G)deno es he leas deg ee o a ai h ul ansi i e
pe mu a ion ep esen a ion o G, ha is, he smalles index o a co e- ee subg oup o G.
De ini ion 1.7. Le Gbe a g oup and le Abe a cha ac e is ically simple g oup. The symbol kA(G)
deno es he la ges numbe such ha Ghas a no mal sec ion ha is he di ec p oduc o non-
F a ini chie ac o s o G ha a e isomo phic (bu no necessa ily G-isomo phic) o A.
This in a ian was only de ined o non-abelian cha ac e is ically simple g oups, bu he e is no
p oblem in ex ending hei de ini ion o elemen a y abelian g oups. The main esul o [12] is he
ollowing one.
Theo em 1.8 ([12, Theo em 1]).The e exis wo absolu e cons an s 0< α < β such ha o e e y
g oup Gwe ha e
α(d(G) + max
A{log kA(G)
log l(A)})< ν(G)< β d(G) + max
A{log kA(G)
log l(A)},
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34 In . J. G oup Theo y, 9 no. 1 (2020) 31-42 A. Balles e -Bolinches, R. Es eban-Rome o, P. Jim´enez-Se al and H. Meng
whe e A uns h ough he non-abelian chie ac o s o Gin a gi en chie se ies o G.
He e kn(G) deno es he maximum o kA(G) whe e A uns o e he non-abelian cha ac e is ically
simple g oups Awi h l(A)≤n. Acco ding o p i a e communica ions wi h he au ho s, he maximum
on he igh -hand side mus be unde s ood o be ze o when Gis soluble, ha is, when Ghas no
non-abelian chie ac o s. This esul is a consequence o a s onge esul .
Theo em 1.9 ([12, Theo em 9.5]).Le Gbe a d-gene a ed g oup. Then
max {d, max
n≥5
log kn(G)
c7log n−4}≤ν(G)≤cd + max
n≥5
log max{1, kn(G)}
log n+ 3,
whe e cand c7a e wo absolu e cons an s.
The lowe bound depends on he ollowing esul .
Lemma 1.10 ([12, Co olla y 9.3]).Le Gbe a g oup. Then mx(G)≥ kn(G)/nc7 o some x≤nc7.
The e was a misp in in he o iginal e sion, whe e he bound mx(G)≥ kn(G) was s a ed ins ead.
This o m o his esul appea s in [13] and so Theo em 1.9 is modi ied he e o he ollowing esul .
Theo em 1.11 ([12, Theo em 9.5]).Le Gbe a d-gene a ed g oup. Then
max {d, max
n≥5
log kn(G)
c7log n−5}≤ν(G)≤cd + max
n≥5
log max{1, kn(G)}
log n+ 3,
whe e cand c7a e wo absolu e cons an s.
The aims o his su ey a e:
(1) o gi e an in e p e a ion o he in a ian kA(G) o a non-abelian cha ac e is ically simple
g oup A,
(2) o imp o e he bounds o mn(G) and, hence, o ν(G), and
(3) o es ima e he alues o he cons an s in [12, Theo em 9.5].
These objec i es ha e been de eloped in dep h in [2]. We e e he eade o his pape o mo e
de ails and p oo s.
2. La ge cha ac e is ically simple sec ions
Recall ha a p imi i e g oup is a g oup wi h a co e- ee maximal subg oup. The ele ance o
p imi i e g oups in he s udy o he g oup s u c u e is due o he ac ha i Mis a maximal subg oup
o a g oup G, hen M/MGis a co e- ee maximal subg oup o G/MGand so G/MGis p imi i e. A
de ailed s udy o p imi i e g oups and he c owns associa ed is p esen ed in [3, Chap e 1]. The
ollowing classical esul o Bae gi es a classi ica ion o he p imi i e g oups.
Theo em 2.1 (Bae , [1], see also [3, Theo em 1.1.7]).Le Gbe a p imi i e g oup and le Ube a
co e- ee maximal subg oup o G. Exac ly one o he ollowing s a emen s holds:
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In . J. G oup Theo y, 9 no. 1 (2020) 31-42 A. Balles e -Bolinches, R. Es eban-Rome o, P. Jim´enez-Se al and H. Meng 35
(1) Soc(G) = Sis a sel -cen alising abelian minimal no mal subg oup o G,G=US and
U∩S= 1.
(2) Soc(G) = Sis a non-abelian minimal no mal subg oup o G,G=US. In his case, CG(S) =
1.
(3) Soc(G) = A×B, whe e Aand Ba e he wo unique minimal no mal subg oups o G,
G=AU =BU and A∩U=B∩U=A∩B= 1. In his case, A= CG(B),B= CG(A),
and A∼
=B∼
=AB ∩Ua e non-abelian.
We say ha a p imi i e g oup is o ype 1, o ype 2, o o ype 3 acco ding o whe he Gsa is ies
one o he s a emen s 1,2, o 3. A maximal subg oup Mo Gis said o be o ype 1, o ype 2, o o
ype 3 when G/MGis a p imi i e g oup o ype 1, o ype 2, o o ype 3, espec i ely. The heo em
o O’Nan and Sco (see [3, Theo em 1.1.52]) gi es a comple e desc ip ion o all p imi i e g oups o
ype 2.
We ha e he ollowing esul .
Theo em 2.2. Le Gbe a p imi i e g oup o ype 2 wi h socle B. Then G/B has no chie ac o s
isomo phic o B.
Wi h he help o Theo em 2.2 and he p ec own associa ed o a supplemen ed chie ac o and a
maximal subg oup supplemen ing i , s udied in [3, Sec ion 1.2], we can p o e he ollowing esul .
Theo em A ([2, Theo em B]).Le Abe a non-abelian chie ac o o a g oup Gand suppose ha
in a gi en chie se ies o G he e a e kchie ac o s isomo phic o A. Then he e exis wo no mal
subg oups Cand Ro Gsuch ha R≤Cand C/R is isomo phic o a di ec p oduc o kminimal
no mal subg oups o G/R isomo phic o A. In pa icula , kA(G)is he numbe o chie ac o s o
Gisomo phic o Ain a gi en chie se ies o G.
An ex ension o his esul o non-F a ini abelian chie ac o s is also p esen ed in [2].
3. New bounds o he numbe o maximal subg oups o a gi en index
We will imp o e he lowe bound in Theo em 1.9 in wo ways: by showing ha he cons an can
be aken o be c7= 2, and by p o ing ha he esul o he o iginal e sion o he pape is co ec
and can e en be imp o ed. By he O’Nan-Sco heo em, we see ha i is enough o p o e he esul
o almos simple g oups. We p o e he ollowing esul .
Theo em 3.1. E e y almos simple g oup Rwi h socle isomo phic o he simple g oup Spossesses a
conjugacy class o co e- ee maximal subg oups o index l(S)o a conjugacy class o co e- ee maximal
subg oups wi h a ixed index s≤l(S)2, depending only on S.
The e o e, c7can be aken as 2. This esul depends on an exhaus i e analysis o he maximal
subg oups o he smalles index o he simple g oups and he associa ed almos simple g oups. As a
consequence, we ob ain he ollowing bound.
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36 In . J. G oup Theo y, 9 no. 1 (2020) 31-42 A. Balles e -Bolinches, R. Es eban-Rome o, P. Jim´enez-Se al and H. Meng
Theo em 3.2. Le Gbe a d-gene a ed g oup. Then
ν(G)≥max {d, max
A
log kA(G)
2 log l(A)−2.63}.
These bounds can be imp o ed in g oups whose non-abelian composi ion ac o s a e in a ce ain
class o g oups o which e e y almos simple g oup wi h socle isomo phic o Shas a conjugacy class
o co e- ee maximal subg oups o index l(S). Fo such g oups, he cons an c7in he denomina o
can be aken o be 1 and he e m −2.63 can be eplaced by −2.5.
We no e ha he lowe bound is use ul only when kA(G) is la ge o some A. Fo ins ance, i
S= Al (5), which belongs o he abo emen ioned class, and G=S , wi h ∈N, acco ding o a
esul o Wiegold [22] we need = 604= 12 960 000 copies o S o ob ain use ul bounds, g ea e han
he numbe o gene a o s 4 + 2 = 6. The esul o Jaikin-Zapi ain and Pybe , e en i we accep ha
c7= 2, equi es 6026 copies o Sin o de o ob ain some use ul esul s (ν(G)≥29).
Ou uppe bound o he numbe o maximal subg oups o index no a g oup will depend on he
ollowing in a ian s. The i s one conce ns he maximal subg oups o ype 1.
De ini ion 3.3. Le Gbe a g oup and le n∈N,n > 1. We deno e by c A
n(G) he numbe o
c owns associa ed o complemen ed abelian chie ac o s o o de no G, ha is, he numbe o
G-isomo phism classes o complemen ed abelian chie ac o s o Go o de n.
I is clea ha c A
n(G) = 0 unless nis a powe o a p ime.
The second in a ian is use ul o ob ain bounds o he numbe o maximal subg oups o ype 2.
De ini ion 3.4. Le n∈N. The symbol k sn(G)deno es he numbe o non-abelian chie ac o s A
in a gi en chie se ies o Gsuch ha he associa ed p imi i e g oup G/CG(A)has a co e- ee maximal
subg oup o index n.
The ollowing in a ian s conce n maximal subg oups o ype 3.
De ini ions 3.5. Le n∈N.
(1) The symbol k on(G)deno es he numbe o non-abelian chie ac o s Ain a gi en chie se ies
o Gsuch ha |A|=n.
(2) The symbol k omn(G)deno es he maximum o he numbe s kA(G) o A unning o e he
isomo phism ypes o non-abelian chie ac o s o Gwi h |A|=n.
We mus no e ha he non-abelian chie ac o s Ao Go o de n all in o a mos wo isomo phism
classes. I ollows ha k on(G)≤2 k omn(G). No e also ha he in a ian s k sn(G), k on(G), and
k omn(G) a e na u ally ela ed o kn(G).
We can use hese in a ian s o gi e new bounds o ν(G).
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In . J. G oup Theo y, 9 no. 1 (2020) 31-42 A. Balles e -Bolinches, R. Es eban-Rome o, P. Jim´enez-Se al and H. Meng 37
Theo em B ([2, Theo em A]).Le Gbe a d-gene a ed non- i ial g oup. Then
max {d, max
A
log kA(G)
2 log l(A)−2.63}≤ν(G)≤η(G),
whe e in he maximum on he le hand side, A uns o e he isomo phism classes o non-abelian
chie ac o s in a gi en chie se ies o Gand η(G)is a unc ion bounded by a linea combina ion o
dand he maxima o lognc A
n(G),logn k sn(G),logn k on(G), and logn k omn(G).
The de e mina ion o ou unc ion η, p esen ed in Theo em 3.12, ollows om es ima ing he
numbe o maximal subg oups o each ype and a gi en index n. Fo he maximal subg oups o
ype 1, we ha e he ollowing esul .
Theo em 3.6. Le Ube a maximal subg oup o ype 1 o a d-gene a ed g oup Gand le n=|G:U|.
Then he numbe o maximal subg oups Mo Gsuch ha Soc(G/MG)is G-isomo phic o Soc(G/UG)
is less han o equal o
nd−n|H1(G/C, A)|
q−1,
whe e A=C/UGis he unique minimal no mal subg oup o G/UGand q=|EndG/C(A)|. In
pa icula , his numbe is less han nd.
As a consequence, we ob ain he ollowing in o ma ion abou he numbe o ype 1 maximal
subg oups.
Co olla y 3.7. The numbe o ype 1 maximal subg oups Mo index n=p o a d-gene a ed g oup
Gis less han o equal o (nd−1)c A
n(G).
The p oo o his esul depends on some a gumen s o Gasch¨u z [11] and Dalla Vol a and Lucchini
[6].
The ollowing esul is use ul o gi e bounds on he numbe o maximal subg oups o Go ype 2
and index n.
Theo em 3.8 (Pybe [21], see [16, Theo em 1.3] o [9, Theo em 21]).The e exis s a cons an bsuch
ha o e e y g oup Gand e e y n≥2,Ghas a mos nbco e- ee maximal subg oups o index n.
In ac , b= 2 will do.
We can apply i o ob ain a bound o he numbe o maximal subg oups o ype 2 and index n.
Theo em 3.9. Le Gbe a g oup and le n∈N. The numbe o maximal subg oups o Go ype 2
and index nis bounded by k sn(G)n2.
Finally, he numbe o maximal subg oups o ype 3 and index nis bounded in he ollowing
heo em.
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38 In . J. G oup Theo y, 9 no. 1 (2020) 31-42 A. Balles e -Bolinches, R. Es eban-Rome o, P. Jim´enez-Se al and H. Meng
Theo em 3.10. Le Gbe a d-gene a ed g oup and le n∈Nwhich is a powe o he o de o a
non-abelian simple g oup. The numbe o maximal subg oups o Go ype 3 and index nis bounded
by
n2min {nd, k omn(G)−1
2} k on(G).
This esul depends on he s udy o he c owns associa ed o non-abelian chie ac o s, since he
minimal no mal subg oups o G/MGa e G-connec ed.
Deno e by T he se o all p ime powe s and by S he se o all powe s o o de s o non-abelian
simple g oups. Then he elemen s o Ta e possible indices o maximal subg oups o ypes 1 and 2,
he elemen s o Sa e possible indices o maximal subg oups o ypes 2 and 3, and he elemen s o
N (S∪T) can appea only as indices o maximal subg oups o ype 2. The e o e we ha e he ollowing
esul .
Theo em 3.11. (1) I n∈T hen
mn(G)≤(nd−1)c A
n(G) + n2 k sn(G)
≤2 max{ndc A
n(G), n2 k sn(G)}.
(2) I n∈S, hen
mn(G)≤n2 k sn(G) + n2min {nd, k om(G)−1
2} k on(G).
≤2n2max { k sn(G),min {nd, k omn(G)−1
2} k on(G)}.
(3) I n /∈S∪T, hen mn(G)≤n2 k sn(G).
As a consequence, we can p ecise he unc ion ηo Theo em B.
Theo em 3.12. Le Gbe a d-gene a ed non- i ial g oup. Then, o
η(G) := max{d+ 2.02 + max{logn2 + lognc A
n(G)},
4.02 + max{logn2 + logn k sn(G)},
4.02 + max{min{d+ logn2,logn k omn(G)}+ logn k on(G)}},
we ha e ha
ν(G)≤η(G).
4. Discussion
In his sec ion, we will show ha Theo em Bimp o es he p e iously known bounds o ν(G).
Fi s o all, we see ha Theo em 1.2 is a consequence o Theo em B. I is a consequence o he ac s
ha in a non-abelian simple g oup, |Ou S| ≤ √|S|and in a p imi i e g oup o ype 3 whose minimal
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no mal subg oups ha e o de n, i(G)≤√n, bo h p o ed in Sec ion 6 o [2]. Theo em 1.3 is also a
consequence o Theo em B.
One difficul y we ind in o de o deduce Theo em 1.9 om Theo em Bis he ac ha he bounds
o Theo em 1.9 depend on some uni e sal cons an s ha a e known o exis , bu whose alues do
no appea in he li e a u e. The analysis o he p oo s in [12] shows ha hese cons an s appea as
linea combina ions o a se o uni e sal cons an s, some o hem wi hou explici alues known o
us. We do his in [2, Sec ion 6].
Fo ins ance, Bo o ik, Pybe , and Shale [4] p esen ed he ollowing es ima ion o he numbe o
isomo phism classes o non-abelian simple subg oups o Sym(n).
Lemma 4.1 ([4]).The numbe g(n)o isomo phism classes o non-abelian simple subg oups o
Sym(n) o n≥5is O(n).
We can p ecise he alue O(n).
Lemma 4.2. The numbe g(n)o isomo phism classes o non-abelian simple g oups o Sym(n) o
n≥5is a mos 4.89n+ 1 141.33.
We can use his esul o p ecise he known bounds on he numbe o isomo phism classes o
minimal no mal subg oups o p imi i e g oups o ype 2 wi h a co e- ee maximal subg oup o a
gi en index.
Lemma 4.3. The numbe s(n)o isomo phism classes o minimal no mal subg oups o p imi i e
g oups o ype 2 wi h a co e- ee maximal subg oup o index nsa is ies he inequali y s(n)≤n1.266.
Since k sn(G)≤s(n), we ob ain ha
logn k sn(G)≤1.266 + lognmax{1, kn(G)}.
No e also ha
lognc A
n(G)≤c6d+ lognmax{1, kn(G)},
logn k on(G)≤log22 + lognmax{1, kn(G)}.
We can w i e in a s onge o m Co olla y 7.3 o [12].
Theo em 4.4 (c . Jaikin-Zapi ain and Pybe [12, Co olla y 7.3]).Le Gbe a d-gene a ed g oup.
Then he e exis s a cons an c6such ha he numbe o i educible G-modules o size nis a mos
nc6dmax{1, kn(G)}.
This esul and he p e ious bounds can be used o ob ain [12, Theo em 9.5] om ou esul s. We
ob ain ha
ν(G)≤(c6+ 1)d+ 3.02 + max lognmax{1, kn(G)}.
We ob ain he ollowing esul .
DOI: h p://dx.doi.o g/10.22108/ijg .2019.114469.1521
