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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017

  • Then
  • A.B
  • =

Discussion on Some
Properties of an
Infinite Non-abelian

=

  • =
  • Where

Group

Ankur Bala#1, Madhuri Kohli#2

#1 M.Sc. , Department of Mathematics, University of
Delhi, Delhi
#2 M.Sc., Department of Mathematics, University of
Delhi, Delhi

( 1 )

India

Abstract: In this Article, we have discussed some of the properties of the infinite non-abelian group of matrices whose entries from integers with non-zero determinant. Such as the number of elements of order 2, number of subgroups of order 2 in this group. Moreover for every finite group , there exists isomorphic to the group . Keywords: Infinite non-abelian group,

  • such that
  • has a subgroup

( 2 )
Now from (1) and (2) one can easily say that is homomorphism.

Notations: GL(n,Z)

nn ) = 1

  • :
  • &

[aij ]nn aij Z.

  • Now consider
  • ,

det(

[aij ]

Theorem 1:

can be embedded in
Clearly

Hence is injective homomorphism. So, by fundamental theorem of isomorphism one can
Proof: If we prove this theorem for then we are done.
,easily conclude that for all m can be embedded in

  • Let
  • us
  • define
  • a
  • mapping

.

  • Such
  • that

Theorem 2:

is non-abelian infinite group

Proof:

Consider
Consider And let H=
Let

B=

  • A
  • =
  • and

Clearly H is subset of And H has infinite elements. Hence Now
.is infinite group.

  • consider two
  • matrices

Such that determinant of A and B is
.
, where
And

ISSN: 2231-5373

http://www.ijmttjournal.org

Page 108

International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017

Proof: Let G be any group and A(G) be the group of all permutations of set G. For any

Then as Hence,

  • , define a map
  • :
  • such that

Hence,

And by a direct application theorem 1, one can conclude that is infinite and non-abelian is non-abelian.
Theorem 3: Number of elements of order 2 in is infinite and hence number of subgroups of order 2 is infinite. is well defined
Clearly is one-one. Also for any Since is onto.
And hence is permutation.

Proof:

Consider
Let K be set of all such permutations Clearly K is subgroup of A(G).

  • Now define a mapping
  • such that

  • Let
  • , and order of A is 2.

Then
Clearly is well-defined and one-one map.

And consider the following equation

Which shows that is homomorphism Obviously, is onto homomorphism is isomorphism.
And hence the theorem.

Theorem 5:

is isomorphic to some subgroup of

GL(n,Z) for all

.

  • Since
  • then:

Proof: Let

be the permutation group on n

  • Case (i):
  • ,

symbols.

Define  : S GL n,Z

such that:

n

Subcase (i): If we take Then

()   nn  Sn

 

Where

is permutation matrix obtained by

 

nn

  • 1
  • 2

n




i.e. if  

then

1 2  n

R



1

Let

R2

Then our case is



 

nn

Rn

Rth

  • has
  • order
  • 2
  • if

is

R

  • Where
  • is
  • row of identity matrix.

i

i

Clearly c has infinite choices.

  • Clearly
  • is a homomorphism.

Number of elements of order 2 in

infinite. And since number of elements of order 2 = Number of subgroups of order 2 in any group.
Number of subgroups of order 2 in infinite. Also by a direct application of theorem 1, one can easily conclude that this theorem is valid for every
Now consider the kernel of this homomorphism.

ker   :()  I

i  i

i

nn

 ker is trivial. Hence the homomorphism is injective. is

Sn

is isomorphic to some subgroup of

GL(n,Z) for all n

Theorem 6: For every finite group , there exists

.value of Theorem 4: Every finite group can be embedded in

  • such that
  • has a subgroup

for some

  • isomorphic to the group
  • .

ISSN: 2231-5373

http://www.ijmttjournal.org

Page 109

International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017

Proof: It is an obvious observation of theorem 4 and theorem 5.

CONCLUSION

is non-abelian infinite group having infinite number of elements of order 2 as well as subgroups of order 2. Also embedded in can be
, Moreover

  • such
  • for every finite group , there exists

  • that
  • has a subgroup isomorphic to the

  • group
  • .

REFERENCES

[1] Contemporary Abstract Algebra, J.A. Gallian, Cengage
Learning, 2016 .ISBN: 1305887859, 9781305887855
[2] Abstract Algebra (3rd Edition), David S. Dummit , Richard
M. Foote, ISBN: 978-0-471-43334-7
[3] On Embedding of Every Finite Group into a Group of
Automorphism IJMTT, Vol.37, Number 2, ISSN: 2231- 5373
[4] A Course in Abstract Algebra, Vijay K. Khanna, S.K.
Bhambri, Second Edition, Vikas Publishing House Pvt Limited, 1999, 070698675X, 9780706986754

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Page 110

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