Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2
The classification of infinite 2-generator groups of nilpotency class 2, up to isomorphism are given as follows: Theorem 1 Let be a 2-generator group of nilpotency class less than or equal to 2 of the form ⋊ , where is an infinite cyclic group and is a p-group. Then G is isomorphic to exactly one gr...
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| Format: | Conference or Workshop Item |
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2007
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| _version_ | 1796855399456964608 |
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| author | Mohd. Ali, Nor Muhainiah Sarmin, Nor Haniza Kappe, Luise-Charlotte |
| author_facet | Mohd. Ali, Nor Muhainiah Sarmin, Nor Haniza Kappe, Luise-Charlotte |
| author_sort | Mohd. Ali, Nor Muhainiah |
| collection | ePrints |
| description | The classification of infinite 2-generator groups of nilpotency class 2, up to isomorphism are given as follows: Theorem 1 Let be a 2-generator group of nilpotency class less than or equal to 2 of the form ⋊ , where is an infinite cyclic group and is a p-group. Then G is isomorphic to exactly one group of the following types: (1.1) ⋊ ,where (1.2) ⋊ , where (1.3) ⋊ ,where (1.4) where The groups in the above list have nilpotency class two precisely for (1.1), (1.2), and (1.3) and are abelian for (1.4). Theorem 2 Let G be an infinite non-abelian 2-generator group of nilpotency class two. Then G is isomorphic to exactly one group of the following types: (2.1) ⋊ where (2.2) ⋊ , where, for , the component is a -group, for and ⋊ is of Type (1.1), (1.2), (1.3) and (1.4) respectively. Let R be the class of infinite 2-generator groups of nilpotency class 2 of Type 2.2. Using their classification and nonabelian tensor squares, the homological functors in R such as the exterior square, the symmetric square and the Schur multiplier are determined in the Composition Theorem as follows: Theorem 3 (Composotion Theorem). Let G be a group of Type 2.2, that is ⋊ , where, for , the componenst are -groups, pi a prime with and ⋊ is of Type (1.1), (1.2), (1.3) and (1.4) respectively, then , (3.1) , (3.2) , (3.3) , (3.4) , (3.5) , (3.6) , (3.7) , (3.8) where T(H) is the torsion subgroup of a group H and Tp(H) denotes the p-torsion subgroup of a group H. |
| first_indexed | 2024-03-05T18:27:59Z |
| format | Conference or Workshop Item |
| id | utm.eprints-14477 |
| institution | Universiti Teknologi Malaysia - ePrints |
| last_indexed | 2024-03-05T18:27:59Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | utm.eprints-144772017-08-08T04:09:04Z http://eprints.utm.my/14477/ Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2 Mohd. Ali, Nor Muhainiah Sarmin, Nor Haniza Kappe, Luise-Charlotte Q Science (General) The classification of infinite 2-generator groups of nilpotency class 2, up to isomorphism are given as follows: Theorem 1 Let be a 2-generator group of nilpotency class less than or equal to 2 of the form ⋊ , where is an infinite cyclic group and is a p-group. Then G is isomorphic to exactly one group of the following types: (1.1) ⋊ ,where (1.2) ⋊ , where (1.3) ⋊ ,where (1.4) where The groups in the above list have nilpotency class two precisely for (1.1), (1.2), and (1.3) and are abelian for (1.4). Theorem 2 Let G be an infinite non-abelian 2-generator group of nilpotency class two. Then G is isomorphic to exactly one group of the following types: (2.1) ⋊ where (2.2) ⋊ , where, for , the component is a -group, for and ⋊ is of Type (1.1), (1.2), (1.3) and (1.4) respectively. Let R be the class of infinite 2-generator groups of nilpotency class 2 of Type 2.2. Using their classification and nonabelian tensor squares, the homological functors in R such as the exterior square, the symmetric square and the Schur multiplier are determined in the Composition Theorem as follows: Theorem 3 (Composotion Theorem). Let G be a group of Type 2.2, that is ⋊ , where, for , the componenst are -groups, pi a prime with and ⋊ is of Type (1.1), (1.2), (1.3) and (1.4) respectively, then , (3.1) , (3.2) , (3.3) , (3.4) , (3.5) , (3.6) , (3.7) , (3.8) where T(H) is the torsion subgroup of a group H and Tp(H) denotes the p-torsion subgroup of a group H. 2007 Conference or Workshop Item PeerReviewed Mohd. Ali, Nor Muhainiah and Sarmin, Nor Haniza and Kappe, Luise-Charlotte (2007) Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2. In: Proceedings of 15th Mathematical Sciencies National Conference (SKSM-15), (2007), 2007, UiTM Shah Alam. |
| spellingShingle | Q Science (General) Mohd. Ali, Nor Muhainiah Sarmin, Nor Haniza Kappe, Luise-Charlotte Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2 |
| title | Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2 |
| title_full | Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2 |
| title_fullStr | Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2 |
| title_full_unstemmed | Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2 |
| title_short | Some homological functors of infinite non-abelian 2-generators groups of nilpotency class 2 |
| title_sort | some homological functors of infinite non abelian 2 generators groups of nilpotency class 2 |
| topic | Q Science (General) |
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