Discussion on Some Properties of an Infinite Non-Abelian Group

International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 <ul style="display: flex;"><li style="flex:1">Then </li><li style="flex:1">A.B </li><li style="flex:1">=</li></ul>Discussion on Some Properties of an Infinite Non-abelian =<ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">Where </li></ul>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, <ul style="display: flex;"><li style="flex:1">such that </li><li style="flex:1">has a subgroup </li></ul>( 2 ) Now from (1) and (2) one can easily say that is homomorphism. Notations: GL(n,Z) nn ) = 1 <ul style="display: flex;"><li style="flex:1">:</li><li style="flex:1">&</li></ul>[aij ]nn aij Z. <ul style="display: flex;"><li style="flex:1">Now consider </li><li style="flex:1">,</li></ul>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 <ul style="display: flex;"><li style="flex:1">Let </li><li style="flex:1">us </li><li style="flex:1">define </li><li style="flex:1">a</li><li style="flex:1">mapping </li></ul>.<ul style="display: flex;"><li style="flex:1">Such </li><li style="flex:1">that </li></ul>Theorem 2: is non-abelian infinite group Proof: Consider Consider And let H= Let B= <ul style="display: flex;"><li style="flex:1">A</li><li style="flex:1">=</li><li style="flex:1">and </li></ul>Clearly H is subset of And H has infinite elements. Hence Now .is infinite group. <ul style="display: flex;"><li style="flex:1">consider two </li><li style="flex:1">matrices </li></ul>Such that determinant of A and B is . , where And ISSN: 2231-5373 <a href="/goto?url=http://www.ijmttjournal.org" target="_blank">http://www.ijmttjournal.org </a>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, <ul style="display: flex;"><li style="flex:1">, define a map </li><li style="flex:1">:</li><li style="flex:1">such that </li></ul>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). <ul style="display: flex;"><li style="flex:1">Now define a mapping </li><li style="flex:1">such that </li></ul><ul style="display: flex;"><li style="flex:1">Let </li><li style="flex:1">, and order of A is 2. </li></ul>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 .<ul style="display: flex;"><li style="flex:1">Since </li><li style="flex:1">then: </li></ul>Proof: Let be the permutation group on n <ul style="display: flex;"><li style="flex:1">Case (i): </li><li style="flex:1">,</li></ul>symbols. Define  : S GL n,Z such that: <ul style="display: flex;"><li style="flex:1"></li><li style="flex:1"></li></ul>nSubcase (i): If we take Then ()   nn  Sn   Where is permutation matrix obtained by   nn <ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">2</li></ul>n i.e. if   then 1 2  n R 1  Let R2 Then our case is <ul style="display: flex;"><li style="flex:1"></li><li style="flex:1"></li></ul>  nn Rn Rth <ul style="display: flex;"><li style="flex:1">has </li><li style="flex:1">order </li><li style="flex:1">2</li><li style="flex:1">if </li></ul>is R<ul style="display: flex;"><li style="flex:1">Where </li><li style="flex:1">is </li><li style="flex:1">row of identity matrix. </li></ul>iiClearly c has infinite choices. <ul style="display: flex;"><li style="flex:1">Clearly </li><li style="flex:1">is a homomorphism. </li></ul>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 nn  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 <ul style="display: flex;"><li style="flex:1">such that </li><li style="flex:1">has a subgroup </li></ul>for some <ul style="display: flex;"><li style="flex:1">isomorphic to the group </li><li style="flex:1">.</li></ul>ISSN: 2231-5373 <a href="/goto?url=http://www.ijmttjournal.org" target="_blank">http://www.ijmttjournal.org </a>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 <ul style="display: flex;"><li style="flex:1">such </li><li style="flex:1">for every finite group , there exists </li></ul><ul style="display: flex;"><li style="flex:1">that </li><li style="flex:1">has a subgroup isomorphic to the </li></ul><ul style="display: flex;"><li style="flex:1">group </li><li style="flex:1">.</li></ul>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 ISSN: 2231-5373 <a href="/goto?url=http://www.ijmttjournal.org" target="_blank">http://www.ijmttjournal.org </a>Page 110

Discussion on Some Properties of an Infinite Non-Abelian Group

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