presentation Homogenous Monomial Groups mahmut

HomogenousMonomialGroups
Mahmut Kuzucuo§lu
Middle East Technical University
Department of Mathematics,
[email protected]
Joint work with B.V. Oliynyk and V. I. Sushchanskyy
29 March-3 April 2016

Dedicated to the memories of
Guido Zappa, Mario Curzio
and
Wolfgang Kappe

Representations of groups
There are three principal types of representations of groups,
each with its particular eld of usefulness, are the following:
(1)Permutation representation of groups.
(2)Monomial representation of groups.
(3)Linear or matrix representation of groups.
Oystein Ore

Representations of groups
Ore continues:
These three types of representations correspond to an
embedding of the group in the following groups:
(1)The symmetric group.
(2)The complete monomial group.
(3)The full linear group.
The symmetric group and the full linear group have been
exhaustively investigated and many of their principal
properties are known.

Representations of groups
A similar study does not seem to exist for complete
monomial group.

Monomial groups
O. Ore;Theory of monomial groups, Trans. Amer.
Math. Soc.51, (1942) 1564.

Monomial groups
Let us recall the denition of a monomial group.

Monomial groups
LetHba an arbitrary group and letn2N:A monomial
permutation overHis a linear transformation
=

x1x2:::xn
h1xi1
h2xi2
:::hnxin

where each variable is changed into some other variable
multiplied by an element ofH.
The elementshi2Hwill be calledfactors or
multipliersin.
The multiplicationhixij
is a formal multiplication satisfying
(hihj)xk=hi(hjxk).

Monomial groups
Ifis another monomial permutation
=

x1x2:::xn
a1xj1
a2xj2
:::anxjn

whereai's are elements ofH, then
=

x1x2:::xn
a1xj1
a2xj2
:::anxjn

x1x2:::xn
h1xi1
h2xi2
:::hnxin

=

x1 x2:::xn
h1ai1
xji
1
h2ai2
xji
2
:::hnain
xjin

Monomial groups
For
=

x1x2:::xn
h1xi1
h2xi2
:::hnxin

1
=

xi1
xi2
:::xin
h
1
1
x1h
1
2
x2:::h
1
nxn

Monomial groups
The setn(H)of all monomial permutations overHof
variablesx1;:::;xnwith multiplication dened as above
forms a group.
This groupn(H)is calledcomplete monomial group
of degreen.

Monomial groups
The set of all monomial permutations of the form
=

x1x2:::xn
xi1
xi2
:::xin

=

1 2:::n
i1i2:::in

forms a subgroupn(f1g)ofn(H)and it is isomorphic to
the symmetric group onnletters.

Monomial groups
The monomial permutations of the form
=

x1x2:::xn
h1x1h2x2:::hnxn

= [h1;h2;:::;hn]
wherehi2Hforms a subgroup ofn(H)which is
isomorphic to the direct productH:::Hofncopies of
H.
Every monomial permutation can be written uniquely as a
product of monomial permutation and a product
(multiplication).

Monomial groups
For example the elementcan be written as

x1 x2: : :xn
h1xi1
h2xi2
: : :hnxin

=

x1: : :xn
xi1
: : :xin

x1 x2: : :xn
h1x1h2x2: : :hnxn

=

x1: : :xn
xi1: : :xin

[h1; : : : ;hn]
Moreover one may observe that, when we take the
conjugate of a multiplication[h1;:::;hn]by a permutation
2n(f1g), we have that the coordinates of the
multiplication is permuted. Indeed

1
[h1;:::;hn]= [h1;:::;hn]

= [h1
;:::;hn
]hence we
have
n(H)

=HoSn

=(H:::H)oSn
The wreath product is the permutational wreath product.

Monomial groups
LetGbe a group which has a subgroupHof indexninG.
Each elementginGpermutes the cosets and there are
factors coming fromH.
Namely

Monomial groups
Hxi:g=Hxig
where
fHxiji=1;:::;ng
is the set of right cosets ofHinGand
fxiji=1;:::;ng
is the set of right coset representatives ofHinG.
Then eachg2Gdetermines a permutation(g)of the
right cosets andxi:g=hi(g)x
i(g)andnelementshi(g)in
H.

Monomial groups
Then the map
G!GL(n;ZH)
g!Diag(h1(g);h2(g);:::;hn(g))(g)

Monomial groups
0
B
B
B
B
B
@
h1(g)
h2(g)
...
hn(g)
1
C
C
C
C
C
A
0
B
B
B
B
B
@
0 1 0 0 0
0 0 1
1
...
0 0 1 0
0 0 0
1
C
C
C
C
C
A
| {z }
denes a monomorphism fromGintoGL(n;ZH)where
ZHis the group ring ofHover the ring of integers.
n(H)is isomorphic to the subgroup of monomial matrices
inGL(n;Z(H)):

Monomial groups
Monomial groups occur also as centralizers of elements in
symmetric groups.
Example.Indeed centralizer of an element say
(12)(34)(56)2S6is
CS6
((12)(34)(56))

=((Z2oS3)

=3(Z2)
Therefore as Ore suggested monomial groups appear
naturally as centralizers of elements in symmetric groups.

Monomial groups
A monomial permutation of the form
=

x1x2:::xk
a1x2a2x3:::akx1

whereai2His called acycle.
is written in the cycle form(a1x2;a2x3;:::;akx1)

Monomial groups
As in the symmetric groups one can write each monomial
permutation as a product of commuting disjoint cycles.
ExampleLetn=5 andH=S3.
=

x1 x2 x3 x4 x5
(123)x3(234)x2(34)x1(134)x5(23)x4

=

x1 x3
(123)x3(34)x1

x2
(234)x2

x4 x5
(134)x5(23)x4

=

(123)x3(34)x1) ((134)x5(23)x4) ((234)x2)

Monomial groups
As we mentioned before each monomial can be written as a
product of disjoint cycles. Now for each cycle
=

x1x2:::xm
c1x2c2x3:::cmx1

= (c1x2;c2x3;:::;cmx1)
whereci2H;them
th
power ofis

m
=

x1x2:::xm
1x12x2:::mxn

where
1=c1c2:::cm;2=c2c3:::cmc1; :::;
m=cmc1:::cm1

Monomial groups
The elementsi2Hare called thedeterminantsof.
As you observed, the determinants are all conjugate as
2=c
1
1
1c1;:::m=c
1
m1
m1cm1;1=c
1
mmcm
So one observes that for each cycle there is a unique
determinant class inH.

Diagonal Embedding of Monomial groups
Letbe the set of sequences consisting of prime numbers.
Let2and= (p1;p2;:::)be a sequence consisting of
not necessarily distinct primespi.
From the given sequencewe may obtain a divisible
sequence(n1;n2;:::ni;:::)where
n1=p1;andni+1=pi+1ni
we have
nijni+1
for alli=1;2;:::.

Diagonal Embedding of Monomial groups
We may embed a complete monomial groupni
(H)
diagonally intoni+1
(H)as follows.
d
pi+1
: ni
(H)!ni+1
(H)

Diagonal Embedding of Monomial groups
Given2ni
(H)where=

x1x2:::xni
h1xj1
h2xj2
:::hni
xjn
i

we dened
pi+1() =

x1 x2: : :xni
jxni+1 xni+2: : :x2ni
j: : :
h1xj1
h2xj2
: : :hni
xjn
i
jh1xni+j1h2xni+j2: : :hni
xni+jn
i
j: : :

: : :xmni+k: : :
: : :hkxmni+j
k
: : :

whereni+1=nipi+1and= (p1;p2;:::pi:::)is a
sequence of not necessarily distinct primes. This embedding
corresponds to strictly diagonal embedding ofni
(H)into
ni+1
(H).

According to the given sequence of primes we continue to
embed
d
pi+1
: ni
(H)!ni+1
(H)
Then we have the following diagram and we obtain direct
systems from the following embeddings
f1g
d
p
1
!n1
(H)
d
p
2
!n2
(H)
d
p
3
!n3
(H)
d
p
4
!:::
f1g
d
p
1
!An1
(H)
d
p
2
!An2
(H)
d
p
3
!An3
(H)
d
p
4
!:::
whereni=ni1pi;i=1;2;3:::andni
(H)is the
complete monomial group onniletters over the groupH
andAni
(H)is the monomial alternating group onniletters
overHandn0=1.

Monomial groups
The direct limit groups obtained from the above
construction are calledhomogenous monomial group
over the groupHand denoted by(H)and homogenous
alternating groupA(H)overHrespectively.
(H)

=
1
[
i=1
ni
(H)

=S()nB
whereBis the base group which is isomorphic to direct
product of the groupH.

Centralizers of elements in(H)
Question 1.Find necessary and sucient condition for
two elements to be conjugate in(H).
Question 2.Find the structure of centralizer of an
element in(H).
Question 3.When two groups1
(H)and2
(H)are
isomorphic.

Centralizers of elements in(H)
For Question 1, by taking the conjugate of a cycle
=

x1xi1
:::xim
c1xi1
c2xi2
:::cmx1

by an element ofn(H)we may write it in the form
=

x1xi1
:::xim
xi1
xi2
:::ax1

wherea2His an element in the determinant class of.
is called anormal formof.Any monomial permutationis similar to a product of
cycles without common variables, so=1:::rwhere
each cycle is in normal form.

Conjugation of elements in(H)
Lemma 1 ( Ore)
The necessary and sucient condition for two monomial
cycles to be conjugate inn(H)is that they shall have the
same length and the same determinant class.
Therefore two monomial permutations are conjugate if and
only if the cycles in their cycle decomposition may be made
correspond in such a manner that corresponding cycles
have he same length and determinant class.

Conjugation of elements
The direct limit group can be written as
(H) =
1
[
i=1
ni
(H):
For an element2(H);we dene the short cycle type
ofas the cycle type ofin the smallestniwhere
2ni
(H).
For the short cycle type, we put an order according to the
length and the determinant class.

Conjugation of elements
By type of a monomial permutationwe have two
variables length of cycle and determinant class.
t() = (a11r1;a12r1;:::a1i1
r1;a21r2;a22r2;:::;a2i2
r2:::;alil
rl)
aijis the representative of conjugacy class inHandriis the
number of cycles of lengthiin the cycle decomposition of
with determinant classaij:

Conjugation of elements
Therefore by using the above lemma, we may state the
following:
Lemma 2
Two elements of(H)are conjugate in(H)if and only
if they have the same cycle type inni
(H)for some ni
dividing.

Steinitz numbers
For the centralizers of elements and isomorphism question,
we now recall Steinitz numbers (supernatural numbers).

Steinitz numbers
Recall that the formal productn=2
r2
3
r3
5
r5
:::of prime
powers with 0rk 1for allkis called aSteinitz
number(supernatural number).
The set of Steinitz numbers form a partially ordered set
with respect to division, namely if=2
r2
3
r3
5
r5
:::and
=2
s2
3
s3
5
s5
:::be two Steinitz numbers, thenjif and
only ifrpspfor all primep.
Moreover they form a lattice if we dene meet and join as
^=2
minfr2;s2g
3
minfr3;s3g
5
minfr5;s5g
:::and
_=2
maxfr2;s2g
3
maxfr3;s3g
5
maxfr5;s5g
::::

Steinitz numbers
For each sequence, we dene a Steinitz number
Char() =2
r2
3
r3
:::p
rp
i
i
:::
whererpi
is the number of times that the primepirepeat in
. If it repeats innitely often, then we writep
1
i
.
For a group(H)obtained from the sequencewe dene
Char((H)) =Char():

Steinitz numbers
For each Steinitz numberwe can dene a homogenous
monomial group(H)and for each homogenous
monomial group(H)we have a Steinitz number.

Centralizers
Since every cycle is conjugate to a cycle in the normal form
and centralizers of conjugate elements are conjugate
(isomorphic), we may assume that we have the cycles in
the normal form.
Moreover as disjoint cycles commute, we nd centralizer of
a cycle of determinant classa2Hrepeatedmtimes, then
centralizer of an arbitrary cycle will be direct product of the
centralizers for each distinct cycle.

Centralizers
First we have only one cycle.
Let=

x1:::xm
x2:::ax1

be a cycle in the normal form
anda2H:
Then the centralizer ofinm(H)is isomorphic to
Ca=CH(a)hi

Centralizers
Assume that the cycle is repeatedstimes with the same
determinant classa2H. Then
() =

x1x2: : :xm
x2x3: : :ax1

xm+1: : :x2m
xm+2: : :axm+1

: : :

x
(s1)m+1: : :xsm
x
(s1)m+2: : :ax
(s1)+1

| {z }
stimes
Then
C
sm(H)()

=(CH(a)hi)oSs

=s(CH(a)hi)
Then by using this we may nd the structure of centralizer
of an arbitrary element in(H).

Centralizer Special case
Theorem 3
Letbe an element of(H)whereis a product of
cycles with the same determinant class a2H of length m
repeated s times and principal beginning ofis contained
inni
(H)where ni=ms. Then
C
(H)()

=Ca(1
(Ca))
where1=
Char()
ni
s.

Theorem 4
Letbe an element of(H)with principal beginning is in
nk
(H)with its normal form=1:::l;where
i=i1:::iri
where for a xed i theijare the normalized
cycles of the same length miand the determinant class ai.
Then the centralizer
C
(H)()

=Ca1
(1
(Ca1
))Ca2
(2
(Ca2
)):::Cal
(l
(Cal
))
where Cai
is the centralizer of a single element
ij2mi
(H):
The group Cai
consists of elements of the form= [ci]
j
i1
where the element cibelongs to the group CH(ai)and
Char(i) =
Char()
nk
ri.

Centralizers
Observe that homogenous monomial group overHbecomes
homogenous symmetric group whenH=f1g.
Therefore our results are compatible with centralizers of
elements in homogenous symmetric groups.
Centralizers of elements in homogenous symmetric groups
is studied and the following is proved:

Centralizers
Theorem 5 (Güven, Kegel, Kuzucuo§lu [1])
Letbe an innite sequence, g2S()and the type of
principal beginning g02Snk
be t(g0) = (r1;r2;:::;rnk
):
Then
C
S()(g)

=
nk
Dr
i=1
Ci(Ci

oS(i))
where Char(i) =
Char()
nk
rifor i=1;:::;nk:
If ri=0;then we assume that corresponding factor isf1g:

Kroshko-Sushchansky studied the diagonal type of
embeddings and they give a complete characterization of
such groups using Steinitz numbers.
Kroshko N. V.; Sushchansky V. I.; Direct Limits of
symmetric and alternating groups with strictly diagonal
embeddings, Arch. Math.71, 173182, (1998).

Theorem 6
(Kuroshko-Sushchansky, 1998)
Two groups S(1)and S(2)are isomorphic if and only if
Char(S(1)) =Char(S(2)).
By using this theorem we prove the following:

Theorem 7
Let H be any nite group. The groups1
(H)and2
(H)
are isomorphic if and only if Char(1) =Char(2)

Güven Ü. B., Kegel O. H., Kuzucuo§lu M.;Centralizers
of subgroups in direct limits of symmetric groups with
strictly diagonal embedding, Comm. in Algebra,43)(6)
1-15 (2015).
A. Kerber;Representations of Permutation Groups I,
Lecture Notes in Mathematics No: 240, Springer
-Verlag, (1971).
B. V. Oliynyk, V. I. Sushchanskii,Imprimitivity systems
and lattices of normal subgroups in D-Hyperoctahedral
groups, Siberian Math. J.55, 132141,(2014).
O. Ore;Theory of monomial groups, Trans. Amer.
Math. Soc.51, (1942) 1564.

Thank You

presentation Homogenous Monomial Groups mahmut

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