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| In mathematical [[finite group theory]], a '''rank 3 permutation group''' [[Group action|acts]] transitively on a set such that the [[stabilizer (group theory)|stabilizer]] of a point has 3 [[orbit (group theory)|orbit]]s. The study of these groups was started by {{harvs|txt|authorlink=Donald Higman|last=Higman|year1=1964|year2=1971}}. Several of the [[sporadic simple group]]s were discovered as rank 3 permutation groups.
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| ==Classification==
| | Instead, this printing process hold up is meant to defecate Photoshop a break option for users World Health Organization are scarce nerve-racking to touch sensation up and publish a pre-existent manikin. There's enough in that respect to lease users pluck and blusher their models, and straightaway Adobe says it'll be dewy-eyed to black and white them also. Photoshop generates a prevue taxonomic category to MakerBot and Shapeways printers Adobe is adding respective features to score printing a poser well-off. |
| The primitive rank 3 permutation groups are all in one of the following classes:
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| *{{harvtxt|Cameron|1981}} classified the ones such that <math>T\times T\le G\le T_0 {\rm wr} Z/2Z</math> where the [[Socle (mathematics)|socle]] ''T'' of ''T''<sub>0</sub>'' is simple, and ''T''<sub>0</sub>'' is a 2-transitive group of degree √''n''.
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| *{{harvtxt|Liebeck|1987}} classified the ones with a regular elementary abelian normal subgroup
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| *{{harvtxt|Bannai|1971}} classified the ones whose socle is a simple alternating group
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| *{{harvtxt|Kantor|Liebler|1982}} classified the ones whose socle is a simple classical group
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| *{{harvtxt|Liebeck|Saxl|1986}} classified the ones whose socle is a simple exceptional or sporadic group.
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| ==Examples==
| | The consolidation is particularly useful for outputting to Shapeways, a made-to-Holy Order 3D-printing divine service. From interior of Photoshop, you'll be capable to project what a printed mannequin should smell similar when it's made with any of Shapeways' materials, from dark sandstone to strong tan. The app will even out estimation how a good deal the publish line will price. Because Photoshop CC is region of [https://www.vocabulary.com/dictionary/Adobe%27s+subscription Adobe's subscription] service, this update comes at no added monetary value to subscribers and leave be made available for download today. |
| If ''G'' is any 4-transitive group acting on a set ''S'', then its action on pairs of elements of ''S'' is a rank 3 permutation group.<ref>The three orbits are: the fixed pair itself; those pairs having one element in common with the fixed pair; and those pairs having no element in common with the fixed pair.</ref> In particular most of the alternating groups, symmetric groups, and [[Mathieu group]]s have 4-transitive actions, and so can be made into rank 3 permutation groups.
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| The projective general linear group acting on lines in a projective space of dimension at least 3 is a rank-3 permutation group.
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| Several [[3-transposition group]]s are rank-3 permutation groups (in the action on transpositions).
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| It is common for the point-stabilizer of a rank-3 permutation group acting on one of the orbits to be a rank-3 permutation group. This gives several "chains" of rank-3 permutation groups, such as the [[Suzuki chain]] and the chain ending with the [[Fischer groups]].
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| Some unusual rank-3 permutation groups (many from {{harv|Liebeck|Saxl|1986}}) are listed below.
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| {| class="wikitable"
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| ! Group !! Point stabilizer !! size !! Comments
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| | G<sub>2</sub>(2) = U<sub>3</sub>(3).2 || PSL<sub>3</sub>(2) || 36 = 1+14+21 || [[Suzuki chain]]
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| | [[Mathieu group|M<sub>11</sub>]] || M<sub>9</sub>.2 || 55 = 1+18+36 || Pairs of points
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| | [[Mathieu group|M<sub>12</sub>]] || M<sub>10</sub>.2 || 66 = 1+20+45 || Pairs of points; two classes
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| | [[Mathieu group|M<sub>22</sub>]] || 2<sup>4</sup>A<sub>6</sub>|| 77 = 1+16+60 ||
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| | [[Janko group J2|J<sub>2</sub>]] || PSU<sub>3</sub>(3)|| 100 = 1+36+63 || [[Suzuki chain]]
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| | [[Higman–Sims group]] || [[Mathieu group|M<sub>22</sub>]] || 100 = 1+22+77
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| | [[Mathieu group|M<sub>22</sub>]] || A<sub>7</sub>|| 176 = 1+70+106 || Two classes
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| | [[Mathieu group|M<sub>23</sub>]] || M<sub>21</sub>.2 || 253 = 1+42+210 || Pairs of points
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| | [[Mathieu group|M<sub>23</sub>]] || 2<sup>4</sup>A<sub>7</sub> || 253 = 1+112+140 ||
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| | [[McLaughlin group]] || U<sub>4</sub>(3) || 275 = 1+112+162
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| | [[Mathieu group|M<sub>24</sub>]] || M<sub>22</sub>.2 || 276 = 1+44+231 || Pairs of points
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| |G<sub>2</sub>(3) || U<sub>3</sub>(3).2 || 351 = 1+126+244 || Two classes
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| ||G<sub>2</sub>(4) || [[Janko group J2|J<sub>2</sub>]] || 416 = 1+100+315 || [[Suzuki chain]]
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| | [[Mathieu group|M<sub>24</sub>]] || M<sub>12</sub>.2 || 1288 = 1+495+792 ||
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| | [[sporadic Suzuki group|Suzuki group]] || G<sub>2</sub>(4) || 1782 = 1+416+1365 || [[Suzuki chain]]
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| ||G<sub>2</sub>(4) || U<sub>3</sub>(4).2 || 2016 = 1+975+1040
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| |[[Conway group|Co<sub>2</sub>]] || PSU<sub>6</sub>(2).2 || 2300 = 1+1891+1408
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| |[[Rudvalis group]] || [[Tits group|²F₄(2)]] || 4060 = 1+1755+2304
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| |[[Fischer group Fi22|Fi<sub>22</sub>]] || 2.PSU<sub>6</sub>(2) || 3510 = 1+693+2816 || 3-transpositions
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| |[[Fischer group Fi22|Fi<sub>22</sub>]] || Ω<sub>7</sub>(3) || 14080 = 1+3159+10920 || Two classes
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| |[[Fischer group Fi23|Fi<sub>23</sub>]] || 2.[[Fischer group|Fi<sub>22</sub>]] || 31671 = 1+3510+28160 ||3-transpositions
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| | G<sub>2</sub>(8) || SU<sub>3</sub>(8) || 130816 = 1+32319+98496
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| |[[Fischer group Fi23|Fi<sub>23</sub>]] || PΩ<sub>8</sub><sup>+</sup>(3).S<sub>3</sub> || 137632 = 1+28431+109200
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| |[[Fischer group Fi24|Fi<sub>24</sub>]]' || [[Fischer group|Fi<sub>23</sub>]] || 306936 = 1+31671+275264 ||3-transpositions
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| |}
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| ==References==
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| {{reflist}}
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| *{{Citation | last1=Bannai | first1=Eiichi | title=Maximal subgroups of low rank of finite symmetric and alternating groups | mr=0357559 | year=72 | month=1971 | journal=Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics | issn=0040-8980 | volume=18 | pages=475–486}}
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| *{{Citation | last1=Brouwer | first1=A. E. | last2=Cohen | first2=A. M. | last3=Neumaier | first3=Arnold | title=Distance-regular graphs | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-50619-5 | mr=1002568 | year=1989 | volume=18}}
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| *{{Citation | last1=Cameron | first1=Peter J. | title=Finite permutation groups and finite simple groups | doi=10.1112/blms/13.1.1 | mr=599634 | year=1981 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=13 | issue=1 | pages=1–22}}
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| *{{Citation | last1=Higman | first1=Donald G. | author1-link=Donald G. Higman | title=Finite permutation groups of rank 3 | doi=10.1007/BF01111335 | mr=0186724 | year=1964 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=86 | pages=145–156}}
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| *{{Citation | last1=Higman | first1=Donald G. | author1-link=Donald G. Higman | title=Actes du Congrès International des Mathématiciens (Nice, 1970) | url=http://mathunion.org/ICM/ICM1970.1/ | publisher=Gauthier-Villars | mr=0427435 | year=1971 | volume=1 | chapter=A survey of some questions and results about rank 3 permutation groups | pages=361–365}}
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| *{{Citation | last1=Kantor | first1=William M. | last2=Liebler | first2=Robert A. | title=The rank 3 permutation representations of the finite classical groups | doi=10.2307/1998750 | mr=648077 | year=1982 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=271 | issue=1 | pages=1–71}}
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| *{{Citation | last1=Liebeck | first1=Martin W. | title=The affine permutation groups of rank three | doi=10.1112/plms/s3-54.3.477 | mr=879395 | year=1987 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=54 | issue=3 | pages=477–516}}
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| *{{Citation | last1=Liebeck | first1=Martin W. | last2=Saxl | first2=Jan | title=The finite primitive permutation groups of rank three | doi=10.1112/blms/18.2.165 | mr=818821 | year=1986 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=18 | issue=2 | pages=165–172}}
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| [[Category:Finite groups]]
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Adobe is stepping into 3D impression today with a novel update to Photoshop CC. It's adding in a fresh 3D-printing prick meant to stimulate it slowly for anyone to pack a model, place or so finish touches on it, and catch it printed. That's on the nose how Adobe sees Photoshop existence victimised for 3D impression as well ?? to a lesser extent as a innovation tool, and more as a manner to place gloss on an existent externalize. Though Photoshop already includes around 3D modelling features, Adobe doesn't look nearly of its users to take up construction objects from the earth up.
Instead, this printing process hold up is meant to defecate Photoshop a break option for users World Health Organization are scarce nerve-racking to touch sensation up and publish a pre-existent manikin. There's enough in that respect to lease users pluck and blusher their models, and straightaway Adobe says it'll be dewy-eyed to black and white them also. Photoshop generates a prevue taxonomic category to MakerBot and Shapeways printers Adobe is adding respective features to score printing a poser well-off.
Users won't sustain to vex around their exemplar dropping apart, because Photoshop volition mechanically engender irregular supports below and close to their posture to earn for sure that it doesn't give way during printing process. It's also partnering with MakerBot and Shapeways so that Photoshop force out automatically beget previews of how a modeling testament look when it's made by any presumption unity of their printers. A printer-taxonomic group preview will besides be available for 3D Systems' Cube printer, and others leave be added in on an ongoing ground.
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