Abstract Root Subgroups and Simple Groups of Lie-Type:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Basel
Birkhäuser Basel
2001
|
| Series: | Monographs in Mathematics
95 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | It was already in 1964 [Fis66] when B. Fischer raised the question: Which finite groups can be generated by a conjugacy class D of involutions, the product of any two of which has order 1, 2 or 37 Such a class D he called a class of 3-tmnspositions of G. This question is quite natural, since the class of transpositions of a symmetric group possesses this property. Namely the order of the product (ij)(kl) is 1, 2 or 3 according as {i,j} n {k,l} consists of 2,0 or 1 element. In fact, if I{i,j} n {k,I}1 = 1 and j = k, then (ij)(kl) is the 3-cycle (ijl). After the preliminary papers [Fis66] and [Fis64] he succeeded in [Fis71J, [Fis69] to classify all finite "nearly" simple groups generated by such a class of 3-transpositions, thereby discovering three new finite simple groups called M(22), M(23) and M(24). But even more important than his classification theorem was the fact that he originated a new method in the study of finite groups, which is called "internal geometric analysis" by D. Gorenstein in his book: Finite Simple Groups, an Introduction to their Classification. In fact D. Gorenstein writes that this method can be regarded as second in importance for the classification of finite simple groups only to the local group-theoretic analysis created by J. Thompson |
| Physical Description: | 1 Online-Ressource (XIII, 389 p) |
| ISBN: | 9783034875943 9783034875967 |
| DOI: | 10.1007/978-3-0348-7594-3 |
Staff View
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| 500 | |a It was already in 1964 [Fis66] when B. Fischer raised the question: Which finite groups can be generated by a conjugacy class D of involutions, the product of any two of which has order 1, 2 or 37 Such a class D he called a class of 3-tmnspositions of G. This question is quite natural, since the class of transpositions of a symmetric group possesses this property. Namely the order of the product (ij)(kl) is 1, 2 or 3 according as {i,j} n {k,l} consists of 2,0 or 1 element. In fact, if I{i,j} n {k,I}1 = 1 and j = k, then (ij)(kl) is the 3-cycle (ijl). After the preliminary papers [Fis66] and [Fis64] he succeeded in [Fis71J, [Fis69] to classify all finite "nearly" simple groups generated by such a class of 3-transpositions, thereby discovering three new finite simple groups called M(22), M(23) and M(24). But even more important than his classification theorem was the fact that he originated a new method in the study of finite groups, which is called "internal geometric analysis" by D. Gorenstein in his book: Finite Simple Groups, an Introduction to their Classification. In fact D. Gorenstein writes that this method can be regarded as second in importance for the classification of finite simple groups only to the local group-theoretic analysis created by J. Thompson | ||
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| isbn | 9783034875943 9783034875967 |
| language | English |
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| spelling | Timmesfeld, Franz Georg Verfasser aut Abstract Root Subgroups and Simple Groups of Lie-Type by Franz Georg Timmesfeld Basel Birkhäuser Basel 2001 1 Online-Ressource (XIII, 389 p) txt rdacontent c rdamedia cr rdacarrier Monographs in Mathematics 95 It was already in 1964 [Fis66] when B. Fischer raised the question: Which finite groups can be generated by a conjugacy class D of involutions, the product of any two of which has order 1, 2 or 37 Such a class D he called a class of 3-tmnspositions of G. This question is quite natural, since the class of transpositions of a symmetric group possesses this property. Namely the order of the product (ij)(kl) is 1, 2 or 3 according as {i,j} n {k,l} consists of 2,0 or 1 element. In fact, if I{i,j} n {k,I}1 = 1 and j = k, then (ij)(kl) is the 3-cycle (ijl). After the preliminary papers [Fis66] and [Fis64] he succeeded in [Fis71J, [Fis69] to classify all finite "nearly" simple groups generated by such a class of 3-transpositions, thereby discovering three new finite simple groups called M(22), M(23) and M(24). But even more important than his classification theorem was the fact that he originated a new method in the study of finite groups, which is called "internal geometric analysis" by D. Gorenstein in his book: Finite Simple Groups, an Introduction to their Classification. In fact D. Gorenstein writes that this method can be regarded as second in importance for the classification of finite simple groups only to the local group-theoretic analysis created by J. Thompson Mathematics Group theory Group Theory and Generalizations Mathematik Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 gnd rswk-swf Untergruppe (DE-588)4224972-7 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 s Untergruppe (DE-588)4224972-7 s 1\p DE-604 Lie-Gruppe (DE-588)4035695-4 s 2\p DE-604 https://doi.org/10.1007/978-3-0348-7594-3 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Timmesfeld, Franz Georg Abstract Root Subgroups and Simple Groups of Lie-Type Mathematics Group theory Group Theory and Generalizations Mathematik Lie-Gruppe (DE-588)4035695-4 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd Untergruppe (DE-588)4224972-7 gnd |
| subject_GND | (DE-588)4035695-4 (DE-588)4295326-1 (DE-588)4224972-7 |
| title | Abstract Root Subgroups and Simple Groups of Lie-Type |
| title_auth | Abstract Root Subgroups and Simple Groups of Lie-Type |
| title_exact_search | Abstract Root Subgroups and Simple Groups of Lie-Type |
| title_full | Abstract Root Subgroups and Simple Groups of Lie-Type by Franz Georg Timmesfeld |
| title_fullStr | Abstract Root Subgroups and Simple Groups of Lie-Type by Franz Georg Timmesfeld |
| title_full_unstemmed | Abstract Root Subgroups and Simple Groups of Lie-Type by Franz Georg Timmesfeld |
| title_short | Abstract Root Subgroups and Simple Groups of Lie-Type |
| title_sort | abstract root subgroups and simple groups of lie type |
| topic | Mathematics Group theory Group Theory and Generalizations Mathematik Lie-Gruppe (DE-588)4035695-4 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd Untergruppe (DE-588)4224972-7 gnd |
| topic_facet | Mathematics Group theory Group Theory and Generalizations Mathematik Lie-Gruppe Lineare algebraische Gruppe Untergruppe |
| url | https://doi.org/10.1007/978-3-0348-7594-3 |
| work_keys_str_mv | AT timmesfeldfranzgeorg abstractrootsubgroupsandsimplegroupsoflietype |