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In the area of modern algebra known as [[group theory]], the '''Janko groups''' are the four [[sporadic simple group]]s ''[[Janko group J1| J<sub>1</sub>]]'', ''[[Janko group J2| J<sub>2</sub>]]'', ''[[Janko group J3| J<sub>3</sub>]]'' and ''[[Janko group J4| J<sub>4</sub>]]'' introduced by [[Zvonimir Janko]]. Unlike the [[Mathieu groups]], [[Conway groups]], or [[Fischer groups]], the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical. |
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== History == |
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Janko constructed the first of these groups, ''J''<sub>1</sub>, in 1965 and predicted the existence of ''J''<sub>2</sub> and ''J''<sub>3</sub>. In 1976, he suggested the existence of ''J''<sub>4</sub>. Later, ''J''<sub>2</sub>, ''J''<sub>3</sub> and ''J''<sub>4</sub> were all shown to exist. |
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''J''<sub>1</sub> was the first sporadic simple group discovered in nearly a century: until then only the [[Mathieu group]]s were known, ''M''<sub>11</sub> and ''M''<sub>12</sub> having been found in 1861, and ''M''<sub>22</sub>, ''M''<sub>23</sub> and ''M''<sub>24</sub> in 1873. The discovery of ''J''<sub>1</sub> caused a great "sensation"<ref>[[Dieter Held]], ''[http://www.mathematik.uni-mainz.de/arbeitsgruppen/gruppentheorie/held/klassifi.pdf Die Klassifikation der endlichen einfachen Gruppen] {{webarchive|url=https://web.archive.org/web/20130626081216/http://www.mathematik.uni-mainz.de/arbeitsgruppen/gruppentheorie/held/klassifi.pdf |date=2013-06-26 }}'' (the classification of the finite simple groups), Forschungsmagazin der [[University of Mainz|Johannes Gutenberg-Universität Mainz]] 1/86 |
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* The [[Janko group J1]] has order {{nowrap|175 560 {{=}} 2<sup>3</sup>{{·}}3{{·}}5{{·}}7{{·}}11{{·}}19}}. It is the only Janko group whose existence was proved by Janko himself. |
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</ref> and "surprise"<ref>The group theorist [[Bertram Huppert]] said of ''J''<sub>1</sub>: |
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* The [[Hall–Janko group]] has order {{nowrap|604 800 {{=}} 2<sup>7</sup>{{·}}3<sup>3</sup>{{·}}5<sup>2</sup>{{·}}7}}. It is also known as ''J''<sub>2</sub>, ''HJ'', or the Hall–Janko–Wales group. It was constructed by [[Marshall Hall (mathematician)|Marshall Hall, Jr.]] and David Wales. |
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"There were a very few things that surprised me in my life... There were only the following two events that really surprised me: the discovery of the first Janko group and the fall of the [[Berlin Wall]]." [http://www.croatianhistory.net/etf/janko/index.html#janko_groups] |
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* The [[Janko group J3]] has order {{nowrap|50 232 960 {{=}} 2<sup>7</sup>{{·}}3<sup>5</sup>{{·}}5{{·}}17{{·}}19}}. It is also known as the Higman–Janko–McKay group. It was constructed by [[Graham Higman]] and [[John McKay]]. |
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</ref> among group theory specialists. This began the modern theory of sporadic groups. |
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* The [[Janko group J4]] has order {{nowrap|86 775 571 046 077 562 880 {{=}} 2<sup>21</sup>{{·}}3<sup>3</sup>{{·}}5{{·}}7{{·}}11<sup>3</sup>{{·}}23{{·}}29{{·}}31{{·}}37{{·}}43}}. It was constructed by [[Simon P. Norton]]. |
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And in a sense, ''J''<sub>4</sub> ended it. It would be the last sporadic group (and, since the non-sporadic families had already been found, the last finite simple group) predicted and discovered, though this could only be said in hindsight when the [[classification of finite simple groups|Classification theorem]] was completed. |
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== References == |
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<references/> |
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{{SIA|mathematics}} |
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[[fr:Groupe de Janko]] |
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[[nl:Janko-groep]] |
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[[zh:揚科群]] |
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Latest revision as of 11:08, 3 September 2024
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In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups J1, J2, J3 and J4 introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway groups, or Fischer groups, the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical.
History
[edit]Janko constructed the first of these groups, J1, in 1965 and predicted the existence of J2 and J3. In 1976, he suggested the existence of J4. Later, J2, J3 and J4 were all shown to exist.
J1 was the first sporadic simple group discovered in nearly a century: until then only the Mathieu groups were known, M11 and M12 having been found in 1861, and M22, M23 and M24 in 1873. The discovery of J1 caused a great "sensation"[1] and "surprise"[2] among group theory specialists. This began the modern theory of sporadic groups.
And in a sense, J4 ended it. It would be the last sporadic group (and, since the non-sporadic families had already been found, the last finite simple group) predicted and discovered, though this could only be said in hindsight when the Classification theorem was completed.
References
[edit]- ^ Dieter Held, Die Klassifikation der endlichen einfachen Gruppen Archived 2013-06-26 at the Wayback Machine (the classification of the finite simple groups), Forschungsmagazin der Johannes Gutenberg-Universität Mainz 1/86
- ^ The group theorist Bertram Huppert said of J1: "There were a very few things that surprised me in my life... There were only the following two events that really surprised me: the discovery of the first Janko group and the fall of the Berlin Wall." [1]
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