In the following we dis uss some re ent results about groups whi h are the produ t of two abelian-by-nite subgroups A and B

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Produ ts of groups whi h ontain abelian

subgroups of nite index

Bernhard Amberg

Fa hberei hPhysik,Mathematikund Informatik,

InstitutfürMathematikderUniversitätMainz,

D-55099Mainz,Germany

amberguni-mainz.de

Abstra t. ItisunknownwhethereverygroupG = AB^whi
hîs^the^produ
tôf^twoâbelian-

by-nitesubgroupsAândB ^mustâlways^haveâ^solubleôrêven^metabelian^subgroupôf^nite

index. Here we deal with the spe ial ase of this problem when A ând B ôntain âbelian

subgroupsof"small"index,notablyofindexatmost2^.^Some^re
ent^results^on^the^solubility

ofsu hgroupsaredis ussedwhi hdependonspe ial al ulationsinvolvinginvolutions.

Keywords:fa torizedgroups,abeliansubgroups,solublegroups,dihedralgroups.

MSC 2010 lassi ation:20D40,20F16,20F50

1 Introdu tion

A group G îs âlled fa torized, if G = AB = {ab | a ∈ A, b ∈ B} îs ^the

produ toftwo subgroupsA ^andB ^ofG^.

If a fa torized group G = AB ^with ^two ^subgroups A ^and B ^is ^given, ^the

questionarisesinwhi hwaythefa torizationasaprodu toftwosubgroupshas

inuen e on the stru ture of the fa torized group. What an be said about G

if thestru tures of the two subgroupsA ând B âre^known? ^Statementsôf ^this

type areof spe ialinterest when they arevalid for arbitrary groups G^without

additional assumptions for G^. ^In ^the ^following ^we ^dis
uss ^some ^re
ent ^results

about groups whi h are the produ t of two abelian-by-nite subgroups A ^and B^.

2 It's Theorem

Thefollowing elebrated theoremofN.It (1955)wasvery inuential inthe

investigation of fa torized groups. It is the basis for almost all known results

aboutprodu ts oftwo abelian subgroups.

Theorem 2.1. If the group G = AB îs ^the ^produ
tôf ^two âbelian ^subgroups A

andB^, ^then G ^ismetabelian.

http://siba-ese.unisalento.it/

²⁰¹⁴^Università^del^Salento

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Theproofisbyasurprisinglyshort ommutator al ulation(seeforinstan e

[1℄, Theorem 2.1.1). It seems almost impossible to generalizethis argument to

more generalsituations, for instan efor produ tsof twonilpotent groups(even

of lasstwo).

3 Produ ts of abelian-by-nite groups

In view of It's theorem the following onje ture has been made (see also

Question 3in[1℄).

Conje ture. Let the group G = AB ^be ^the ^produ
t ^of ^two ^abelian-

by-nite subgroups A ând B ^(i.e. A ând B ^have âbelian ^subgroups ôf

niteindex).ThenG^issoluble-by-niteandperhapsevenmetabelian- by-nite.

The validityofthis onje ture ouldonlybevariedsofar underadditional

requirements. Ya.Sysak proved it in 1986 for linear groups (see [19℄ and [20℄),

and J.Wilsonin1990for residually nitegroups (see[8℄ or [1℄,Theorem 2.3.4).

The importan e of the following theorem of N.Chernikov (1981) lies inthe

fa t that it requires no additional assumptions on the fa torized group G ^and

uses onlyproperties fromthefa torization ofG^(see ^[1℄,^Theorem ^2.2.5).

t ôf ^two entral-by-nite subgroups A ând B^, ^then G îssoluble-by-nite.

It isunknownwhetherG ^must^be metabelian-by-nite inthis ase.

IfN îsâ ^normal^subgroupôfâ^groupGândA ândB âreâbelian^subgroups

of G^, ^then ^subgroup N A = N A∩ AB = A(NA ∩ B) ^is ^metabelian ^by ^It's

theorem.

This useful extension of It's theorem maybe generalized to any subgroup

H ^as^follows ^(see^[20℄, ^Lemma^9).

Lemma 3.2. LetG ^be â ^group ândA ândB ^two âbelian ^subgroups ôf G^. Îf ^the

subgroup H ôf G îs ôntained ⁱⁿ ^the ^setAB^, ^then H îs metabelian.

ConsidernowagroupG = AB^whi
hîs^the^produ
tôf^two^subgroupsAând B^,^whi
h ^have ^(abelian)^subgroup A₀ ^resp.B₀ ôf^nite îndex n =| A : A0 |ând m =| B : B0 |^.^Then ^by ^Lemma^1.2.5 ôf ^[1℄ ^the^subgroup < A₀, B₀ > ^has^nite

index at most nmⁱⁿG^.

Clearly,if alsowe should have that A₀B₀ = B₀A₀ îs â ^subgroupôf G^,^then G^hasâ ^metabelian^subgroup ôf ^niteîndex ^(byÎt).

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This shows that if additional permutability onditions are imposed, some

fa torizationproblems be ome mu heasier andsometimes trivial.

Todealwiththeabove onje tureitisnaturalto onsiderrstthe asewhen

the indi es of the abelian subgroups A0 ⁱⁿ A ând B0 ⁱⁿ B âre ^small. ^Wê ^will

onsiderthefollowing

Problem. Let the group G = AB ^be ^the ^produ
t ôf ^two ^subgroups A ândB^, ^whereA ôntains ân âbelian ^subgroup A₀ ând B ôntains ân

abeliansubgroupB₀ ^su
h ^that^the^indi
esn =| A : A0|^andm =| B : B0 |

are at most2. Is then G ^soluble ^and/or metabelian-by-nite?

4 Finiteprodu tsof subgroupswith solublesubgroups

of index at most 2

Finite produ ts of two subgroups that ontain y li subgroups of index

at most 2 ^were ^for înstan
e ônsidered ^by ^B.Huppert ^[11℄, ^W.S
ott ^[17℄ ând

V.Monakhov[14℄and[15℄.

V.Monakhovshowed in[14℄that anite groupG = AB îs ^solubleîf Aând B ^have ^y
li^subgroups ôfîndex ât ^most 2^.

ThefollowingresultbyL.Kazarin[13℄generalizesthewell-knowntheoremof

O.KegelandH.Wielandtonthesolubilityofeveryniteprodu toftwonilpotent

subgroups(seefor instan e[1℄, Theorem2.4.3).

Theorem 4.1. If the nite group G = AB ^is ^the ^produ
t ^of ^two ^subgroups A

andB^, êa
h ôf ^whi
h ^possesses â ^nilpotent ^subgroup ôf îndex ât^most 2^, ^then G

issoluble.

5 Produ ts of y li -by-nite groups

Produ ts of nite y li groups were for instan e studied in [10℄ (see also

[12℄). P.Cohn proved in [9℄ that every group G = AB ^whi
h ^is ^the^produ
t ^of

two innite y li subgroups A ^and B ^has ^a non-trivialnormal subgroup of G

whi h is ontained in A ôr B^. ^This împlies ^that G ^has â ^subgroup H ôf ^nite

index whose derived subgroup H^′ ^and ^the ^fa
tor ^group H/H^′ ^are ^both ^y
li

(see [18℄), so thatG = AB ^is meta y li -by-nite.The question arises whether every group whi h is the produ t of two y li -by-nite subgroups, must be

meta y li -by-nite.

The following theoremof B.Amberg and Ya.Sysak [5℄ generalizesthe result

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t ^of ^two ^subgroups A ^and B^,

ea hof whi hhas a y li subgroup ofindex atmost2^,^then G^ismeta y li -by-

nite.

Note that a non-abelian innite group whi h has a y li group of index 2

mustbetheinnitedihedralgroup. Thisensurestheexisten eof involutionsin

A ^andB ^whi
h ^we ^may^use^for omputationsinsidethefa torized groupG^.

An important idea in the proof is to show that the normalizer in G ^of ^an

innite y li subgroupofoneofthefa torsA^orB ^has^anon-trivialinterse tion withtheother fa tor.

Re all that a group G îs ^dihedral îf ît ân ^be ^generated ^by ^two ^distin
t

involutions. The stru ture of su h groups is well-known. The main properties

are olle ted inthefollowing lemma.

Lemma5.2. LetthedihedralgroupG^be ^generated^by^the^two involutionsx^and y^.^Letc = xy ^and C =< c >^. ^Then^we ^have

a) The y li subgroup C ^is^normal ⁱⁿ G ^with ^index 2^,^the ^group G = C⋊ <

i > îs^the ^semidire
t ^produ
tôf C ând â ^subgroup < i > ôf ôrder^2,

b) If G ^isnon-abelian, then C ^is hara teristi in G^.

) Everyelementof G r C îsânînvolution^whi
hînvertsêveryêlement ôfC^,

i.e. if g∈ G r C^, ^then c^g = c⁻¹ ^for c∈ C^,

d) The set G r C îs â ^single ônjuga
y ^lass îf ând ônly îf ^the ôrder ôf C îs

nite andodd;it is the union of two onjuga y lasses otherwise.

6 Finite Produ ts of dihedral groups

By Theorem 4.1 nite produ ts of dihedral subgroupsare soluble. The fol-

lowing more pre i estatement abouttheir derived length isproved in[2℄.

Theorem 6.1. Let G = AB ^be â ^nite ^group, ^whi
h îs â ^produ
t ôf ^subgroups A ândB^, ^where A îs ^dihedral ând B îs êither ^y
li ôr â ^dihedral ^group. ^Then G⁽⁷⁾= 1^.

Itisalsoshownin[2℄thatif Gîsâ^nite2^-group^then^weêven^haveG⁽⁵⁾ = 1

in this ase, but there exist nite 2^-groups ^of ^derived ^lenth 3 ^whi
h ^are ^the

produ tof twodihedral subgroups.

TheproofofTheorem6.1isbasedonmethodsfornitefa torizedgroups.We

mentionhereonly twowell-knownlemmas thatplayaroleinourinvestigations

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Lemma 6.2. Let the nite group G = AB ^be ^the ^produ
t ^of ^two ^subgroups A

and B^, ^then ^for êvery ^prime p ^there êxistsâ ^Sylow-p^-subgroup ôf G ^whi
h îs â

produ t of a Sylow-p^-sub^group ôf A ând â ^Sylow-p^-subgroup ôf B^.

Lemma 6.3. Let the nite group G = AB ^be ^the ^produ
t ôf ^subgroups A ând B ând ^let A₀ ând B₀ ^be ^normal ^subgroups ôf A ândB^, respe tively. If A₀B₀ = B0A0^, ^then A^x₀B0 = B0A^x₀ ^for âl^l x ∈ G^. Âssume ⁱⁿ âddition ^that A0 ând B0

are π^-groups ^for â ^setôf ^primesπ^. Îf O_π(G) = 1^, ^then [A^G₀, B₀^G] = 1^.

7 Produ ts of periodi lo ally dihedral groups

AgroupGîs^lo
ally^dihedralîfît^hasâ^lo
al^systemôf^dihedral^subgroups,

i.e.every nitesubset of Gîsôntained ⁱⁿ^some ^dihedral^subgroup ôfG^.

Every periodi lo ally dihedral group is lo ally nite and every nite

subgroupof su ha groupis ontained ina nitedihedral subgroup.

Lemma 7.1. Every periodi lo ally dihedral group G^has ^a ^lo
ally ^yli ^normal

subgroup C ôf îndex 2^,ând êveryêlement ôf G r C îsân învolution^that înverts

every element of C^;

G = C⋊ < i >îs ^the ^semidire
t ^produ
tôf C ând â ^subgroup < i > ôf ôrder^2.

The following well-knownlemma isuseful for thestudy oflo allynite fa -

torizedgroups(see for instan e[1℄,Lemma1.2.3).

Lemma7.2. Letthelo allynitegroupG = AB ^be^the^produ
tôf^two^subgroups AândB^,ând^letA₀ ândB₀ ^be ^nite^normal^subgroupsôfA ândB^,respe tively.

Thenthere exists a nite subgroup E ^of G ^su
h ^that A0, B0⊆ E ⊆ NG(A0, B0) ^andE = (A∩ E)(B ∩ E)^.

Thefollowingsolubility riterionisprovedin[2℄).Thespe ial asewhenthe

groupG^is^periodi ^was^already^dealt ^withⁱⁿ^[4℄.

Theorem 7.3. Let the group G = AB ^be ^the ^produ
t ^of ^two ^periodi ^lo
al^ly

dihedral subgroups A ^andB^. ^Then G^is ^soluble.

For a omplete proof we refer the reader to [2℄. Here we sket h only the

argument that if the result is false, there will be a ounterexample with no

nontrivial solublenormalsubgroup.

AssumethatthetheoremisfalseandthereexistsanonsolublegroupG = AB

withperiodi lo allydihedralsubgroupsA ^andB^.^Then A = A₀hci^,B = B₀hdi

fortwo involutionsc∈ A r A0 ^and d∈ B r B0^,^withcac = a⁻¹ ^for ^ea
h a∈ A0

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and dbd = b⁻¹ ^for êa
hb ∈ B0^; A₀ ând B₀ âre ^lo
ally ^y
li ^normal^subgroups

of A^resp.B^.

By Lemma3.2it iseasy to see thatbothsubgroupsA ^and B ^must^be^non-

abelian. The ase that A∩ B =< c > îs êasier ând ônsidered ^rst. ^Thus ^we

mayassumethat A∩ B = 1^.

Assume that N 6= 1îs â ^soluble^normal ^subgroup ôf G^.^Clearly R = N A = A(N A∩ B) îs ^soluble ^by ^the ^modular ^law, ând ^so ^lo
ally ^nite ^(as â ^soluble

produ tof twoperiodi groups).

If L îs â ^nite ^normal ^subgroup ôf A ând S îs â ^nite ^normal ^subgroup

of N A∩ B^, ^then ^the ^subgroup H =< L, S > îs ^nite ând ^so ^the ^normalizer K = NR(< L, S >) = (K∩ A)(K ∩ B) îs^fa
torized ^by^Lemma^7.2.

Sin ethenitegroupK/CK(H)îs^the^produ
tôf^two^subgroupsôf^dihedral

groups,itsderivedlengthisatmost7byTheorem6.1.Sin eH∩C^K(H) = Z(H)^,

this impliesthatH⁽⁸⁾ = 1.

If R₀ =< A₀, R∩ B0>^,^thenR⁽⁸⁾₀ = 1 ând^sin
e | R : R0 |≤ 4^,^the^subgroup Rând^soN ^have^derived^lengthât^most^9.^Thenâlso^the^produ
tT ôfâll^soluble

normal subgroupsof Gîs â ^soluble^normal ^subgroup ôf Gôf ^derived ^length ât

most9.ThusG/T ^is^a ounterexamplewithnosolublenormalsubgroupN 6= 1^.

The laim isproved.

8 Produ ts of generalized dihedral groups

It turns outthatgroups withthefollowing property an be handled byour

methods.

Denition8.1. AgroupGîsgeneralizeddihedralifitisofdihedraltype, i.e. G ôntains ân âbelian ^subgroup X ôf îndex ² ând ân învolution τ ^whi
h

inverts every element inX^.

It iseasy toseethatA = X⋊ < a >îs ^thesemi-dire tprodu tof anabelian subgroup X ândân învolutiona^,^so^thatxâ= x⁻¹ ^for êa
hx∈ X^.

Clearly dihedraland lo allydihedral groupsarealso generalizeddihedral.

The main propertiesof generalizeddihedral groups are olle ted inthe fol-

lowing lemma.

Lemma 8.2. Let A ^be ^a generalized dihedral group. Thenthe following holds 1) every subgroup of X ^is ^normalⁱⁿ A^;

2) ifA ^isnon-abelian, thenevery non-abelian normalsubgroup of A^ontains

the derived subgroup A^′ ^of A^;

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3) A^′ = X² ând ^so ^the ômmutator ^fa
tor ^group A/A^′ îs ân êlementary

abelian 2^-group;

4) the enter of A ôin
ides ^with ^the ^set ôf âll involutions of X^;

5) the oset aX ôin
ides ^with ^the ^setôf âl^l non- entral involutions of A^;

6) two involutions aând b ⁱⁿ A âre ônjugateîf ând ônly îf ab⁻¹ ∈ X²^.

7) if A ^is non-abelian, then X ^is hara teristi in A^.

Thesolubilityofeveryprodu tof twogeneralizeddihedralgroups isproved

inB.Amberg andYa.Sysak [6℄.

t ôf ^two ^subgroups A ând B^, êa
h ôf ^whi
h îsêither âbêlian ôr generalized dihedral. Then Gîs ^soluble.

Theproof ofthis theoremiselementary anduses almostonly omputations

withinvolutions.Extensiveuseismadebythefa tthateverytwoinvolutionsof

agroupgeneratea dihedralsubgroup.A mainideaof theproofis toshowthat

thenormalizerinGôf â non-trivialnormalsubgroupof one ofthefa torsA ôr B ^hasâ non-trivialinterse tionwiththeother fa tor.

If this is not the ase we may nd ommuting involutions inA ^and B ^and

produ ea nontrivial abelian normalsubgroup byother omputations. Thefol-

lowing lemma gives some onditions under whi h two permutable generalized

dihedralsubgroupsAând B ôf â^group^have ^permutableinvolutionsa∈ Aând b ∈ B^. Â ^spe
ial âse ôf ^this îs âlready ûsed ⁱⁿ ^the ^proof ôf ^Theorem ^5.1 ând

dis ussedin[20℄.

Lemma 8.4. Let G ^be â ^group ôf ^the ^form G = AB ^with ^subgroups A ând B

su hthat A = X⋊ < c >ândB = Y ⋊ < d > ^forâbelian^subgroupsX ândY ând

involutions când d^.Îf x^c= x⁻¹ ^for êa
h x∈ X ând N_A(< y >) = 1 = N_B(< x >)

foreverynon-trivial y li normalsubgroup< x > ôfA ând< y >ôf B^,^then^the

subgroup B ^is non-abelian and there exist involutions cx ∈ A ^and yd ∈ B ^su
h

that (cx)(yd) = (yd)(cx)^.

The ase when one of thetwo subgroupsA ând B îs âbelian, îs ônsidered

separatelyand leads to strongerresults. In this asewe obtain a bound on the

solubilitylength ofG^.

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t ^of ^a ^generalize^d ^dihedral

subgroup A ând ân âbelian ^subgroup B^, ^then ^the ^derived ^length ôf G ^does ^not

ex eed 5^.

Thefollowing onsequen eofTheorem8.3shouldbe omparedwithTheorem

5.1.

Corollary 8.6. Let the group G = AB ^be ^the ^produ
t ôf ^two ^subgroups A ând B^, êa
h ôf ^whi
hôntains âtorsion-free lo ally y li subgroup of index atmost 2. ThenG îs ^solubleând metabelian-by-nite.

9 Groups saturated by dihedral subgroups

A groupGîs^saturated ^by ^subgroupsⁱⁿâ ^setSîfêvery^nite^subgroup S ôf Gîs ôntainedⁱⁿ^subgroup ôfG ^whi
hîs îsomorphi^to â^subgroup ⁱⁿS^.

Lemma 9.1. A lo ally nite group whi h is saturated by dihedral subgroups is

lo allydihedral.

Proof. Let x ând y ^be ^two êlements ôf G ^with o(x) > 2 ând o(y) > 2^. ^By

hypothesis the nite group < x, y > îs ôntained ⁱⁿ â ^proper ^nite ^dihedral

groupD =< a > ⋊ < i >^.^Sin
e x∈< a >^,y ∈< a >^,^it ^follows ^thatxy = yx^.

This shows that the elements of G ^with ^order ^more ^than ² ^generate ^a ^lo
ally

y li normalsubgroupHôfG^.^Clearly^the^setG r H îs^non-emptyândônsists

only ofinvolutions.

Let t∈ G r H ^be â ^xed ând x∈ G r H ân ârbitrary involution. If h∈ H

with o(x) > 2^, ^then ^the ^nite ^subgroup < h, x, t > îs ôntained ⁱⁿ â ^dihedral

subgroup D =< h1 > ⋊ < t >^. ^Then h1 ∈ H ^by ^the ^denition ôf H^. ^Thus x∈ D ⊆ H⋊ < t > ^for êvery învolution x∈ G^. Ît^follows ^thatG = H⋊ < t >^.

Thelemma isproved.

QED

A.ShlopkinandA.Rubashkinin[16℄extendedLemma9.1toseveral lassesof

periodi groups.UsingTheorem7.3onprodu tsoftwoperiodi lo allydihedral

subgroupswe prove thefollowing in[4℄.

Theorem 9.2. If the innite periodi group G ^is ^saturated ^by ^nite ^dihedral

subgroups, then G ^is^a ^lo
ally ^nite ^dihedral ^group.

Assume there exists a periodi group G ^saturated ^by ^dihedral ^subgroups

whi h is not lo ally dihedral. Then G ^is ^not ^lo
ally ^nite ^by ^Lemma ^9.1. ^By

results in [16℄ the entralizer CG(γ) ôf êvery învolution γ ⁱⁿ G îs â ^(nite ôr

innite)periodi lo ally dihedralgroup,andthere existatleasttwo involutions

τ ând µ6= τ ⁱⁿâ Sylow-2-subgroupS ôf G^.

In the following we dis uss some re ent results about groups whi h are the produ t of two abelian-by-nite subgroups A and B

In the following we dis uss some re ent results about groups whi h are the produ t of two abelian-by-nite subgroups A and B