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Discussion on Some Properties of an Infinite Non-Abelian Group International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 Discussion on Some Then A.B = Properties of an Infinite Non-abelian = Group Where = 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 such that has a subgroup isomorphic to the group . ( 2 ) Keywords: Infinite non-abelian group, Now from (1) and (2) one can easily say that is Notations: GL(,) n Z []aij n n : aij Z. & homomorphism. det( ) = 1 Now consider , Theorem 1: can be embedded in Clearly Proof: If we prove this theorem for , Hence is injective homomorphism. then we are done. So, by fundamental theorem of isomorphism one can Let us define a mapping easily conclude that can be embedded in for all m . Such that Theorem 2: is non-abelian infinite group Proof: Consider Consider Let A = and And let H= Clearly H is subset of . And H has infinite elements. B= Hence is infinite group. Now 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. Hence, is non-abelian. For any , define a map : such that And by a direct application theorem 1, one can conclude that is infinite and non-abelian Then as Theorem 3: Number of elements of order 2 in Hence, is well defined is infinite and hence number of Clearly is one-one. subgroups of order 2 is infinite. Proof: Consider Also for any Since is onto. And hence is permutation. 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 for all . Since then: GL(,) n Z Proof: Let be the permutation group on n Case (i): , symbols. Define :,Sn GL n Z such that: Subcase (i): If we take () S nn n Then Where is permutation matrix obtained by nn 12 n i.e. if then 12 n R1 Let R2 Then our case is nn has order 2 if Rn th Where Ri is R row of identity matrix. Clearly c has infinite choices. i Clearly is a homomorphism. Number of elements of order 2 in is Now consider the kernel of this homomorphism. infinite. And since number of elements of order 2 = Number ker : ( ) Inn i i i of subgroups of order 2 in any group. ker is trivial. Number of subgroups of order 2 in is Hence the homomorphism is injective. infinite. Also by a direct application of theorem 1, one can Sn is isomorphic to some subgroup of easily conclude that this theorem is valid for every for all n . value of Theorem 6: For every finite group , there exists 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 can be embedded in , Moreover for every finite group , there exists such 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 ISSN: 2231-5373 http://www.ijmttjournal.org Page 110 .
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