38:
817:
3122:
for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized
2917:
2842:
2625:
2708:
2244:
of the group of
Euclidean translations with a group of origin-preserving transformations, and this product structure is respected by the inclusion of the Euclidean group in the affine group. This gives,
2006:
2997:
In this case there are points for which the set of images under the isometries is not closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of
3428:
reflection with respect to a plane, and a translation in that plane, a rotation about an axis perpendicular to the plane, or a reflection with respect to a perpendicular plane
1815:
2174:
2083:
1946:
1848:
1787:
1745:
1522:
1446:
2422:
2470:
483:
458:
421:
1913:
2541:
2940:
They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: O
3465:
In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class.
2847:
3478:
Rotations about an axis combined with translation along that axis are in the same class if the angle is the same and the translation distance is the same.
2773:
3497:
3096:
all isometries which are a combination of a rotation about some axis and a proportional translation along the axis; in general this is combined with
2646:
1562:
are among the oldest and most studied, at least in the cases of dimension 2 and 3 â implicitly, long before the concept of group was invented.
3487:
Rotations about an axis by the same angle not equal to 180°, combined with reflection in a plane perpendicular to that axis, are in the same class.
1234:
1663:). They include the translations and rotations, and combinations thereof; including the identity transformation, but excluding any reflections.
3440:
rotation by 180° about an axis and rotation by 180° about a perpendicular axis (results in rotation by 180° about the axis perpendicular to both)
785:
1282:
1287:
3137:
1576:
1277:
1272:
2565:
3093:
a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction
1092:
1356:
1239:
343:
3552:
3399:
3568:. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp.
3512:
1387:
293:
778:
288:
3573:
2386:
1539:, depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the
2437:
3388:
1678:, such as a reflection about some hyperplane, every other indirect isometry can be obtained by the composition of
3010:(e.g., in 2D all translations in one direction, and all translations by rational distances in another direction).
1249:
2383:) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:
2176:, which describes the initial position of the body. The position and orientation of the body at any later time
1951:
1555:
or
Euclidean motions. They comprise arbitrary combinations of translations and rotations, but not reflections.
704:
2510:
3593:
3517:
1244:
1224:
771:
3007:
Non-countable groups, where there are points for which the set of images under the isometries is not closed
1449:
1189:
1097:
3484:
Reflections in a plane combined with translation in that plane by the same distance are in the same class.
2200:) for any later time. For that reason, the direct Euclidean isometries are also called "rigid motions".
388:
202:
1792:
1229:
120:
2150:
2059:
1922:
1824:
1763:
1721:
1498:
1422:
1640:
3115:; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a
3502:
3407:
3270:
2752:
2717:
1492:
1457:
1380:
864:
586:
320:
197:
85:
3087:
ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis
3019:
all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the
466:
441:
404:
2335:. All affine theorems apply. The origin of Euclidean geometry allows definition of the notion of
2141:
1644:
1865:
1528:
of the space itself, and contains the group of symmetries of any figure (subset) of that space.
3544:
2364:
1613:
1488:
1484:
1184:
1147:
1115:
1102:
736:
526:
3090:
ditto combined with discrete translation along the axis or with all isometries along the axis
2357:
1216:
884:
610:
3536:
2961:
Countably infinite groups without arbitrarily small translations, rotations, or combinations
959:
949:
939:
929:
3020:
3013:
Non-countable groups, where for all points the set of images under the isometries is closed
2760:
2505:
844:
834:
550:
538:
156:
90:
2983:
translations in independent directions, and possibly a finite point group). This includes
8:
3588:
2994:
Countably infinite groups with arbitrarily small translations, rotations, or combinations
2725:
2129:
1699:
1628:
1559:
1409:
1373:
1361:
1202:
1032:
125:
20:
2331:, the geometry of the Euclidean group of symmetries, is, therefore, a specialisation of
3392:
3325:
2551:
2328:
2324:
2241:
1453:
1133:
1123:
110:
82:
3051:-dimensional subspace combined with a discrete group of isometries in the orthogonal (
3569:
3548:
3537:
3522:
3363:
2764:
2429:
2279:
2209:
1524:; and arbitrary finite combinations of them. The Euclidean group can be seen as the
1197:
1160:
515:
358:
252:
3507:
1312:
1050:
681:
3561:
3255:
3027:
2372:
1332:
1012:
1004:
996:
988:
980:
913:
894:
854:
666:
658:
650:
642:
634:
622:
562:
502:
492:
334:
276:
151:
2117:) as a whole is not connected: there is no continuous trajectory that starts in E(
3456:
2984:
2489:
2332:
1417:
1317:
1055:
826:
750:
743:
729:
686:
497:
327:
241:
181:
61:
3124:
2965:
2632:
1525:
1337:
1155:
1060:
757:
693:
383:
363:
300:
265:
186:
176:
161:
146:
100:
77:
3026:
all isometries that keep the origin fixed, or more generally, some point (the
2964:
i.e., for every point the set of images under the isometries is topologically
1322:
3582:
3004:, and, in 2D, the group generated by a rotation about the origin by 1 radian.
2298:
2287:
1045:
874:
676:
598:
432:
305:
171:
2934:
2309:
2233:
2033:) is continuous. Such a function is called a "continuous trajectory" in E(
1552:
1342:
1327:
1128:
1110:
1040:
531:
230:
219:
166:
141:
136:
95:
66:
29:
3437:
rotation by 180° about an axis and reflection in a plane through that axis
2048:) is connected in this topology. That is, given any two direct isometries
3431:
glide reflection with respect to a plane, and a translation in that plane
3403:
3349:
2988:
2320:
1401:
1168:
1084:
808:
3443:
two rotoreflections about the same axis, with respect to the same plane
2351:
2133:
1307:
1173:
1065:
698:
426:
2912:{\displaystyle {\text{E}}^{+}(n)={\text{SO}}(n)\ltimes {\text{T}}(n).}
2213:
1682:
with some direct isometry. Therefore, the indirect isometries are a
804:
519:
2316:
Details for the first representation are given in the next section.
3131:
3066:-dimensional subspace combined with another one in the orthogonal (
2837:{\displaystyle {\text{SO}}(n)\cong {\text{E}}^{+}(n)/{\text{T}}(n)}
2748:
2336:
2249:, two ways of writing elements in an explicit notation. These are:
2217:
1715:
1648:
1413:
56:
3459:; the translation group is the union of those for all distances.
2434:
or the same orthogonal transformation followed by a translation:
1264:
398:
312:
2755:(in dimensions 2 and 3, these are the familiar reflections in a
2620:{\displaystyle {\text{E}}(n)={\text{T}}(n)\rtimes {\text{O}}(n)}
2756:
2132:, because they describe the physically possible movements of a
1655:), called the special Euclidean group and usually denoted by E(
37:
3418:
For some isometry pairs composition does not depend on order:
2958:
are even maximal among the groups including the next category.
2703:{\displaystyle {\text{O}}(n)\cong {\text{E}}(n)/{\text{T}}(n)}
2128:
The continuous trajectories in E(3) play an important role in
3455:
The translations by a given distance in any direction form a
3434:
inversion in a point and any isometry keeping the point fixed
3112:
2340:
1683:
3472:
Inversions with respect to all points are in the same class.
816:
3107:); the set of images of a point under the isometries is a
3136:
E(1), E(2), and E(3) can be categorized as follows, with
1817:) is defined to converge if and only if, for any point
1639:
The direct isometries (i.e., isometries preserving the
2850:
2776:
2751:, rather than a translation followed by some kind of
2649:
2568:
2513:
2440:
2389:
2153:
2062:
1954:
1925:
1868:
1827:
1795:
1766:
1724:
1501:
1425:
469:
444:
407:
3446:
two glide reflections with respect to the same plane
2987:. Examples more general than those are the discrete
2747:
They are represented as a translation followed by a
2352:
Subgroup structure, matrix and vector representation
2220:
notions can be adapted immediately to this setting.
3498:Fixed points of isometry groups in Euclidean space
3475:Rotations by the same angle are in the same class.
2911:
2836:
2702:
2619:
2535:
2464:
2416:
2356:The Euclidean group is a subgroup of the group of
2168:
2109:. The same is true for the indirect isometries E(
2077:
2000:
1940:
1907:
1842:
1809:
1781:
1739:
1666:The isometries that reverse handedness are called
1516:
1440:
477:
452:
415:
3100:-fold rotational isometries about the same axis (
2759:line or plane, which may be taken to include the
2136:in three-dimensional space over time. One takes
2040:It turns out that the special Euclidean group SE(
1634:
3580:
3132:Overview of isometries in up to three dimensions
1862:From this definition it follows that a function
1235:Representation theory of semisimple Lie algebras
3566:Three-dimensional geometry and topology. Vol. 1
2223:
3462:In 1D, all reflections are in the same class.
2740:isometries. In these cases the determinant of
2240:dimensions. Both groups have a structure as a
1747:implies a topology for the Euclidean group E(
1381:
779:
3481:Reflections in a plane are in the same class
1915:is continuous if and only if, for any point
1460:). The group depends only on the dimension
3425:two rotations or screws about the same axis
2001:{\displaystyle f_{p}:\to \mathbb {E} ^{n}}
1388:
1374:
1273:Particle physics and representation theory
815:
786:
772:
3534:
3400:3D isometries that leave the origin fixed
2156:
2065:
1988:
1928:
1830:
1803:
1769:
1727:
1504:
1428:
471:
446:
409:
2180:will be described by the transformation
1709:
1464:of the space, and is commonly denoted E(
3413:
1240:Representations of classical Lie groups
3581:
3391:asserts that any element of E(3) is a
2770:This relation is commonly written as:
2346:
344:Classification of finite simple groups
2192:is in E(3), the same must be true of
3513:Coordinate rotations and reflections
3450:
2736:), also of index two, consisting of
1456:between any two points (also called
1093:Lie groupâLie algebra correspondence
2631:) is (in the natural way) also the
2546:Together, these facts imply that E(
2113:). On the other hand, the group E(
2085:, there is a continuous trajectory
1694:). It follows that the subgroup E(
13:
3084:all rotations about one fixed axis
2208:The Euclidean groups are not only
1674:. For any fixed indirect isometry
14:
3605:
1810:{\displaystyle i\in \mathbb {N} }
1570:
798:Isometry group of Euclidean space
3080:Examples in 3D of combinations:
2203:
2169:{\displaystyle \mathbb {E} ^{3}}
2078:{\displaystyle \mathbb {E} ^{n}}
1941:{\displaystyle \mathbb {E} ^{n}}
1843:{\displaystyle \mathbb {E} ^{n}}
1782:{\displaystyle \mathbb {E} ^{n}}
1740:{\displaystyle \mathbb {E} ^{n}}
1517:{\displaystyle \mathbb {E} ^{n}}
1474:inhomogeneous special orthogonal
1452:of that space that preserve the
1441:{\displaystyle \mathbb {E} ^{n}}
36:
2417:{\displaystyle x\mapsto A(x+b)}
1612:can be attributed to available
2903:
2897:
2886:
2880:
2869:
2863:
2831:
2825:
2812:
2806:
2788:
2782:
2697:
2691:
2678:
2672:
2661:
2655:
2614:
2608:
2597:
2591:
2580:
2574:
2465:{\displaystyle x\mapsto Ax+c,}
2444:
2411:
2399:
2393:
1983:
1980:
1968:
1902:
1896:
1890:
1887:
1875:
1635:Direct and indirect isometries
1288:Galilean group representations
1283:Poincaré group representations
705:Infinite dimensional Lie group
1:
3539:A Course in Modern Geometries
3535:Cederberg, Judith N. (2001).
3528:
3518:Reflection through the origin
2327:, we read off from this that
1690:), which can be denoted by E(
1551:), whose elements are called
1278:Lorentz group representations
1245:Theorem of the highest weight
3374:
3361:
3347:
3336:
3323:
3312:
3301:
3290:
3253:
3242:
3231:
3220:
3209:
3180:
3169:
3158:
3040:the whole Euclidean group E(
2921:
2224:Relation to the affine group
1531:A Euclidean isometry can be
478:{\displaystyle \mathbb {Z} }
453:{\displaystyle \mathbb {Z} }
416:{\displaystyle \mathbb {Z} }
7:
3491:
1565:
203:List of group theory topics
10:
3610:
3268:
3062:one of these groups in an
3047:one of these groups in an
1908:{\displaystyle f:\to E(n)}
1230:Lie algebra representation
3119:-fold helix of such axes.
2496:): for every translation
1850:, the sequence of points
1458:Euclidean transformations
3503:Euclidean plane isometry
3271:Euclidean plane isometry
3033:all direct isometries E(
2926:Types of subgroups of E(
2718:special orthogonal group
2543:is again a translation.
2536:{\displaystyle u^{-1}tu}
2363:It has as subgroups the
1594:, which gives 3 in case
1225:Lie group representation
321:Elementary abelian group
198:Glossary of group theory
3287:Preserves orientation?
3206:Preserves orientation?
3155:Preserves orientation?
2562:), which is written as
2308:, as explained for the
2232:) is a subgroup of the
2142:identity transformation
1751:). Namely, a sequence
1541:special Euclidean group
1479:The Euclidean group E(
1250:BorelâWeilâBott theorem
3314:Rotation about an axis
3233:Rotation about a point
2913:
2838:
2704:
2621:
2537:
2466:
2418:
2358:affine transformations
2228:The Euclidean group E(
2170:
2079:
2002:
1942:
1909:
1844:
1811:
1783:
1741:
1614:translational symmetry
1518:
1442:
1148:Semisimple Lie algebra
1103:Adjoint representation
737:Linear algebraic group
479:
454:
417:
3338:Reflection in a plane
3182:Reflection in a point
2979:a group generated by
2914:
2839:
2720:, is a subgroup of O(
2705:
2622:
2538:
2467:
2419:
2343:can then be deduced.
2171:
2080:
2003:
1943:
1910:
1845:
1812:
1784:
1742:
1710:Topology of the group
1519:
1443:
1217:Representation theory
480:
455:
418:
3594:Euclidean symmetries
3414:Commuting isometries
3376:Inversion in a point
3244:Reflection in a line
2848:
2774:
2647:
2627:. In other words, O(
2566:
2511:
2438:
2387:
2379:). Any element of E(
2151:
2060:
1952:
1923:
1866:
1825:
1793:
1764:
1722:
1647:subsets) comprise a
1616:, and the remaining
1499:
1423:
467:
442:
405:
3284:Degrees of freedom
3277:
3276:Isometries of E(3)
3203:Degrees of freedom
3196:
3195:Isometries of E(2)
3152:Degrees of freedom
3145:
3144:Isometries of E(1)
3074:)-dimensional space
3059:)-dimensional space
2732:) has a subgroup E(
2500:and every isometry
2347:Detailed discussion
2130:classical mechanics
1718:of Euclidean space
1629:rotational symmetry
1543:, often denoted SE(
1362:Table of Lie groups
1203:Compact Lie algebra
111:Group homomorphisms
21:Algebraic structure
3393:screw displacement
3326:Screw displacement
3275:
3194:
3143:
3138:degrees of freedom
2909:
2844:or, equivalently:
2834:
2728:two. Therefore, E(
2700:
2617:
2552:semidirect product
2533:
2462:
2414:
2329:Euclidean geometry
2325:Erlangen programme
2242:semidirect product
2210:topological groups
2166:
2075:
1998:
1938:
1905:
1840:
1807:
1779:
1737:
1577:degrees of freedom
1514:
1454:Euclidean distance
1438:
1134:Affine Lie algebra
1124:Simple Lie algebra
865:Special orthogonal
587:Special orthogonal
475:
450:
413:
294:Lagrange's theorem
3554:978-0-387-98972-3
3523:Plane of rotation
3451:Conjugacy classes
3386:
3385:
3364:Improper rotation
3281:Type of isometry
3267:
3266:
3200:Type of isometry
3192:
3191:
3149:Type of isometry
2895:
2878:
2855:
2823:
2798:
2780:
2689:
2670:
2653:
2606:
2589:
2572:
2430:orthogonal matrix
2280:orthogonal matrix
1760:of isometries of
1398:
1397:
1198:Split Lie algebra
1161:Cartan subalgebra
1023:
1022:
914:Simple Lie groups
796:
795:
371:
370:
253:Alternating group
210:
209:
3601:
3562:William Thurston
3558:
3542:
3422:two translations
3389:Chasles' theorem
3278:
3274:
3256:Glide reflection
3197:
3193:
3146:
3142:
3106:
3028:orthogonal group
3003:
3002:
2978:
2918:
2916:
2915:
2910:
2896:
2893:
2879:
2876:
2862:
2861:
2856:
2853:
2843:
2841:
2840:
2835:
2824:
2821:
2819:
2805:
2804:
2799:
2796:
2781:
2778:
2709:
2707:
2706:
2701:
2690:
2687:
2685:
2671:
2668:
2654:
2651:
2626:
2624:
2623:
2618:
2607:
2604:
2590:
2587:
2573:
2570:
2558:) extended by T(
2542:
2540:
2539:
2534:
2526:
2525:
2481:
2471:
2469:
2468:
2463:
2423:
2421:
2420:
2415:
2373:orthogonal group
2319:In the terms of
2307:
2278:
2264:
2175:
2173:
2172:
2167:
2165:
2164:
2159:
2121:) and ends in E(
2084:
2082:
2081:
2076:
2074:
2073:
2068:
2007:
2005:
2004:
1999:
1997:
1996:
1991:
1964:
1963:
1947:
1945:
1944:
1939:
1937:
1936:
1931:
1914:
1912:
1911:
1906:
1849:
1847:
1846:
1841:
1839:
1838:
1833:
1816:
1814:
1813:
1808:
1806:
1788:
1786:
1785:
1780:
1778:
1777:
1772:
1746:
1744:
1743:
1738:
1736:
1735:
1730:
1626:
1607:
1600:
1593:
1523:
1521:
1520:
1515:
1513:
1512:
1507:
1483:) comprises all
1447:
1445:
1444:
1439:
1437:
1436:
1431:
1390:
1383:
1376:
1333:Claude Chevalley
1190:Complexification
1033:Other Lie groups
919:
918:
827:Classical groups
819:
801:
800:
788:
781:
774:
730:Algebraic groups
503:Hyperbolic group
493:Arithmetic group
484:
482:
481:
476:
474:
459:
457:
456:
451:
449:
422:
420:
419:
414:
412:
335:Schur multiplier
289:Cauchy's theorem
277:Quaternion group
225:
224:
51:
50:
40:
27:
16:
15:
3609:
3608:
3604:
3603:
3602:
3600:
3599:
3598:
3579:
3578:
3555:
3531:
3494:
3457:conjugacy class
3453:
3416:
3273:
3134:
3101:
3000:
2998:
2969:
2957:
2951:
2945:
2924:
2892:
2875:
2857:
2852:
2851:
2849:
2846:
2845:
2820:
2815:
2800:
2795:
2794:
2777:
2775:
2772:
2771:
2686:
2681:
2667:
2650:
2648:
2645:
2644:
2603:
2586:
2569:
2567:
2564:
2563:
2518:
2514:
2512:
2509:
2508:
2490:normal subgroup
2473:
2439:
2436:
2435:
2388:
2385:
2384:
2354:
2349:
2333:affine geometry
2302:
2270:
2254:
2226:
2206:
2160:
2155:
2154:
2152:
2149:
2148:
2140:(0) to be the
2069:
2064:
2063:
2061:
2058:
2057:
2016:
1992:
1987:
1986:
1959:
1955:
1953:
1950:
1949:
1948:, the function
1932:
1927:
1926:
1924:
1921:
1920:
1867:
1864:
1863:
1858:
1834:
1829:
1828:
1826:
1823:
1822:
1802:
1794:
1791:
1790:
1773:
1768:
1767:
1765:
1762:
1761:
1759:
1731:
1726:
1725:
1723:
1720:
1719:
1712:
1637:
1617:
1602:
1595:
1584:
1573:
1568:
1508:
1503:
1502:
1500:
1497:
1496:
1450:transformations
1448:; that is, the
1432:
1427:
1426:
1424:
1421:
1420:
1418:Euclidean space
1412:of (Euclidean)
1406:Euclidean group
1394:
1349:
1348:
1347:
1318:Wilhelm Killing
1302:
1294:
1293:
1292:
1267:
1256:
1255:
1254:
1219:
1209:
1208:
1207:
1194:
1178:
1156:Dynkin diagrams
1150:
1140:
1139:
1138:
1120:
1098:Exponential map
1087:
1077:
1076:
1075:
1056:Conformal group
1035:
1025:
1024:
1016:
1008:
1000:
992:
984:
965:
955:
945:
935:
916:
906:
905:
904:
885:Special unitary
829:
799:
792:
763:
762:
751:Abelian variety
744:Reductive group
732:
722:
721:
720:
719:
670:
662:
654:
646:
638:
611:Special unitary
522:
508:
507:
489:
488:
470:
468:
465:
464:
445:
443:
440:
439:
408:
406:
403:
402:
394:
393:
384:Discrete groups
373:
372:
328:Frobenius group
273:
260:
249:
242:Symmetric group
238:
222:
212:
211:
62:Normal subgroup
48:
28:
19:
12:
11:
5:
3607:
3597:
3596:
3591:
3577:
3576:
3559:
3553:
3530:
3527:
3526:
3525:
3520:
3515:
3510:
3508:Poincaré group
3505:
3500:
3493:
3490:
3489:
3488:
3485:
3482:
3479:
3476:
3473:
3452:
3449:
3448:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3415:
3412:
3384:
3383:
3380:
3377:
3373:
3372:
3369:
3366:
3360:
3359:
3356:
3353:
3346:
3345:
3342:
3339:
3335:
3334:
3331:
3328:
3322:
3321:
3318:
3315:
3311:
3310:
3307:
3304:
3300:
3299:
3296:
3293:
3289:
3288:
3285:
3282:
3265:
3264:
3261:
3258:
3252:
3251:
3248:
3245:
3241:
3240:
3237:
3234:
3230:
3229:
3226:
3223:
3219:
3218:
3215:
3212:
3208:
3207:
3204:
3201:
3190:
3189:
3186:
3183:
3179:
3178:
3175:
3172:
3168:
3167:
3164:
3161:
3157:
3156:
3153:
3150:
3133:
3130:
3129:
3128:
3125:dihedral group
3120:
3094:
3091:
3088:
3085:
3078:
3077:
3076:
3075:
3060:
3045:
3038:
3031:
3024:
3021:rotation group
3014:
3011:
3008:
3005:
2995:
2992:
2962:
2959:
2953:
2952:. The groups I
2947:
2941:
2938:
2923:
2920:
2908:
2905:
2902:
2899:
2891:
2888:
2885:
2882:
2874:
2871:
2868:
2865:
2860:
2833:
2830:
2827:
2818:
2814:
2811:
2808:
2803:
2793:
2790:
2787:
2784:
2765:rotoreflection
2763:, or in 3D, a
2699:
2696:
2693:
2684:
2680:
2677:
2674:
2666:
2663:
2660:
2657:
2633:quotient group
2616:
2613:
2610:
2602:
2599:
2596:
2593:
2585:
2582:
2579:
2576:
2532:
2529:
2524:
2521:
2517:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2353:
2350:
2348:
2345:
2314:
2313:
2295:
2225:
2222:
2205:
2202:
2163:
2158:
2072:
2067:
2012:
1995:
1990:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1962:
1958:
1935:
1930:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1854:
1837:
1832:
1805:
1801:
1798:
1776:
1771:
1755:
1734:
1729:
1711:
1708:
1636:
1633:
1575:The number of
1572:
1571:Dimensionality
1569:
1567:
1564:
1526:symmetry group
1511:
1506:
1435:
1430:
1396:
1395:
1393:
1392:
1385:
1378:
1370:
1367:
1366:
1365:
1364:
1359:
1351:
1350:
1346:
1345:
1340:
1338:Harish-Chandra
1335:
1330:
1325:
1320:
1315:
1313:Henri Poincaré
1310:
1304:
1303:
1300:
1299:
1296:
1295:
1291:
1290:
1285:
1280:
1275:
1269:
1268:
1263:Lie groups in
1262:
1261:
1258:
1257:
1253:
1252:
1247:
1242:
1237:
1232:
1227:
1221:
1220:
1215:
1214:
1211:
1210:
1206:
1205:
1200:
1195:
1193:
1192:
1187:
1181:
1179:
1177:
1176:
1171:
1165:
1163:
1158:
1152:
1151:
1146:
1145:
1142:
1141:
1137:
1136:
1131:
1126:
1121:
1119:
1118:
1113:
1107:
1105:
1100:
1095:
1089:
1088:
1083:
1082:
1079:
1078:
1074:
1073:
1068:
1063:
1061:Diffeomorphism
1058:
1053:
1048:
1043:
1037:
1036:
1031:
1030:
1027:
1026:
1021:
1020:
1019:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
975:
974:
970:
969:
968:
967:
961:
957:
951:
947:
941:
937:
931:
924:
923:
917:
912:
911:
908:
907:
903:
902:
892:
882:
872:
862:
852:
845:Special linear
842:
835:General linear
831:
830:
825:
824:
821:
820:
812:
811:
797:
794:
793:
791:
790:
783:
776:
768:
765:
764:
761:
760:
758:Elliptic curve
754:
753:
747:
746:
740:
739:
733:
728:
727:
724:
723:
718:
717:
714:
711:
707:
703:
702:
701:
696:
694:Diffeomorphism
690:
689:
684:
679:
673:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
631:
630:
619:
618:
607:
606:
595:
594:
583:
582:
571:
570:
559:
558:
551:Special linear
547:
546:
539:General linear
535:
534:
529:
523:
514:
513:
510:
509:
506:
505:
500:
495:
487:
486:
473:
461:
448:
435:
433:Modular groups
431:
430:
429:
424:
411:
395:
392:
391:
386:
380:
379:
378:
375:
374:
369:
368:
367:
366:
361:
356:
353:
347:
346:
340:
339:
338:
337:
331:
330:
324:
323:
318:
309:
308:
306:Hall's theorem
303:
301:Sylow theorems
297:
296:
291:
283:
282:
281:
280:
274:
269:
266:Dihedral group
262:
261:
256:
250:
245:
239:
234:
223:
218:
217:
214:
213:
208:
207:
206:
205:
200:
192:
191:
190:
189:
184:
179:
174:
169:
164:
159:
157:multiplicative
154:
149:
144:
139:
131:
130:
129:
128:
123:
115:
114:
106:
105:
104:
103:
101:Wreath product
98:
93:
88:
86:direct product
80:
78:Quotient group
72:
71:
70:
69:
64:
59:
49:
46:
45:
42:
41:
33:
32:
9:
6:
4:
3:
2:
3606:
3595:
3592:
3590:
3587:
3586:
3584:
3575:
3574:0-691-08304-5
3571:
3567:
3563:
3560:
3556:
3550:
3546:
3541:
3540:
3533:
3532:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3495:
3486:
3483:
3480:
3477:
3474:
3471:
3470:
3469:
3466:
3463:
3460:
3458:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3420:
3419:
3411:
3409:
3405:
3401:
3396:
3394:
3390:
3381:
3378:
3375:
3370:
3367:
3365:
3362:
3357:
3354:
3351:
3348:
3343:
3340:
3337:
3332:
3329:
3327:
3324:
3319:
3316:
3313:
3308:
3305:
3302:
3297:
3294:
3291:
3286:
3283:
3280:
3279:
3272:
3262:
3259:
3257:
3254:
3249:
3246:
3243:
3238:
3235:
3232:
3227:
3224:
3221:
3216:
3213:
3210:
3205:
3202:
3199:
3198:
3187:
3184:
3181:
3176:
3173:
3170:
3165:
3162:
3159:
3154:
3151:
3148:
3147:
3141:
3139:
3127:of R, Dih(R).
3126:
3121:
3118:
3114:
3110:
3104:
3099:
3095:
3092:
3089:
3086:
3083:
3082:
3081:
3073:
3069:
3065:
3061:
3058:
3054:
3050:
3046:
3043:
3039:
3036:
3032:
3029:
3025:
3022:
3018:
3017:
3015:
3012:
3009:
3006:
2996:
2993:
2990:
2986:
2982:
2977:
2973:
2967:
2963:
2960:
2956:
2950:
2944:
2939:
2936:
2935:Finite groups
2933:
2932:
2931:
2929:
2919:
2906:
2900:
2889:
2883:
2872:
2866:
2858:
2828:
2816:
2809:
2801:
2791:
2785:
2768:
2766:
2762:
2758:
2754:
2750:
2745:
2743:
2739:
2735:
2731:
2727:
2723:
2719:
2715:
2710:
2694:
2682:
2675:
2664:
2658:
2642:
2638:
2634:
2630:
2611:
2600:
2594:
2583:
2577:
2561:
2557:
2553:
2549:
2544:
2530:
2527:
2522:
2519:
2515:
2507:
2503:
2499:
2495:
2491:
2487:
2482:
2480:
2476:
2459:
2456:
2453:
2450:
2447:
2441:
2432:
2431:
2427:
2408:
2405:
2402:
2396:
2390:
2382:
2378:
2374:
2370:
2366:
2365:translational
2361:
2359:
2344:
2342:
2339:, from which
2338:
2334:
2330:
2326:
2322:
2317:
2311:
2305:
2300:
2299:square matrix
2296:
2293:
2289:
2288:column vector
2285:
2281:
2277:
2273:
2268:
2262:
2258:
2252:
2251:
2250:
2248:
2243:
2239:
2235:
2231:
2221:
2219:
2215:
2211:
2204:Lie structure
2201:
2199:
2195:
2191:
2187:
2183:
2179:
2161:
2146:
2143:
2139:
2135:
2131:
2126:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2070:
2055:
2051:
2047:
2043:
2038:
2036:
2032:
2028:
2024:
2020:
2015:
2011:
1993:
1977:
1974:
1971:
1965:
1960:
1956:
1933:
1918:
1899:
1893:
1884:
1881:
1878:
1872:
1869:
1860:
1857:
1853:
1835:
1820:
1799:
1796:
1774:
1758:
1754:
1750:
1732:
1717:
1707:
1705:
1701:
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1664:
1662:
1658:
1654:
1650:
1646:
1642:
1632:
1630:
1624:
1620:
1615:
1611:
1605:
1598:
1591:
1587:
1582:
1578:
1563:
1561:
1556:
1554:
1553:rigid motions
1550:
1546:
1542:
1538:
1534:
1529:
1527:
1509:
1494:
1490:
1486:
1482:
1477:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1433:
1419:
1415:
1411:
1407:
1403:
1391:
1386:
1384:
1379:
1377:
1372:
1371:
1369:
1368:
1363:
1360:
1358:
1355:
1354:
1353:
1352:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1306:
1305:
1298:
1297:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1270:
1266:
1260:
1259:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1222:
1218:
1213:
1212:
1204:
1201:
1199:
1196:
1191:
1188:
1186:
1183:
1182:
1180:
1175:
1172:
1170:
1167:
1166:
1164:
1162:
1159:
1157:
1154:
1153:
1149:
1144:
1143:
1135:
1132:
1130:
1127:
1125:
1122:
1117:
1114:
1112:
1109:
1108:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1090:
1086:
1081:
1080:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1038:
1034:
1029:
1028:
1017:
1011:
1009:
1003:
1001:
995:
993:
987:
985:
979:
978:
977:
976:
972:
971:
966:
964:
958:
956:
954:
948:
946:
944:
938:
936:
934:
928:
927:
926:
925:
921:
920:
915:
910:
909:
900:
896:
893:
890:
886:
883:
880:
876:
873:
870:
866:
863:
860:
856:
853:
850:
846:
843:
840:
836:
833:
832:
828:
823:
822:
818:
814:
813:
810:
806:
803:
802:
789:
784:
782:
777:
775:
770:
769:
767:
766:
759:
756:
755:
752:
749:
748:
745:
742:
741:
738:
735:
734:
731:
726:
725:
715:
712:
709:
708:
706:
700:
697:
695:
692:
691:
688:
685:
683:
680:
678:
675:
674:
671:
665:
663:
657:
655:
649:
647:
641:
639:
633:
632:
628:
624:
621:
620:
616:
612:
609:
608:
604:
600:
597:
596:
592:
588:
585:
584:
580:
576:
573:
572:
568:
564:
561:
560:
556:
552:
549:
548:
544:
540:
537:
536:
533:
530:
528:
525:
524:
521:
517:
512:
511:
504:
501:
499:
496:
494:
491:
490:
462:
437:
436:
434:
428:
425:
400:
397:
396:
390:
387:
385:
382:
381:
377:
376:
365:
362:
360:
357:
354:
351:
350:
349:
348:
345:
342:
341:
336:
333:
332:
329:
326:
325:
322:
319:
317:
315:
311:
310:
307:
304:
302:
299:
298:
295:
292:
290:
287:
286:
285:
284:
278:
275:
272:
267:
264:
263:
259:
254:
251:
248:
243:
240:
237:
232:
229:
228:
227:
226:
221:
220:Finite groups
216:
215:
204:
201:
199:
196:
195:
194:
193:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
143:
140:
138:
135:
134:
133:
132:
127:
124:
122:
119:
118:
117:
116:
113:
112:
108:
107:
102:
99:
97:
94:
92:
89:
87:
84:
81:
79:
76:
75:
74:
73:
68:
65:
63:
60:
58:
55:
54:
53:
52:
47:Basic notions
44:
43:
39:
35:
34:
31:
26:
22:
18:
17:
3565:
3538:
3467:
3464:
3461:
3454:
3417:
3397:
3387:
3135:
3116:
3108:
3102:
3097:
3079:
3071:
3067:
3063:
3056:
3052:
3048:
3041:
3034:
2989:space groups
2980:
2975:
2971:
2954:
2948:
2942:
2927:
2925:
2769:
2746:
2741:
2737:
2733:
2729:
2721:
2713:
2711:
2640:
2636:
2628:
2559:
2555:
2547:
2545:
2501:
2497:
2493:
2485:
2483:
2478:
2474:
2433:
2425:
2380:
2376:
2368:
2362:
2355:
2318:
2315:
2310:affine group
2303:
2297:by a single
2291:
2283:
2275:
2271:
2266:
2260:
2256:
2246:
2237:
2234:affine group
2229:
2227:
2207:
2197:
2193:
2189:
2185:
2184:(t). Since
2181:
2177:
2144:
2137:
2127:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2094:
2093:) such that
2090:
2086:
2053:
2049:
2045:
2041:
2039:
2034:
2030:
2026:
2022:
2018:
2013:
2009:
1916:
1861:
1855:
1851:
1818:
1756:
1752:
1748:
1714:The natural
1713:
1703:
1695:
1691:
1687:
1679:
1675:
1671:
1667:
1665:
1660:
1656:
1652:
1638:
1622:
1618:
1609:
1608:. Of these,
1603:
1601:, and 6 for
1596:
1589:
1585:
1580:
1574:
1557:
1548:
1544:
1540:
1536:
1532:
1530:
1485:translations
1480:
1478:
1473:
1469:
1465:
1461:
1405:
1399:
1343:Armand Borel
1328:Hermann Weyl
1129:Loop algebra
1111:Killing form
1085:Lie algebras
1070:
962:
952:
942:
932:
898:
888:
878:
868:
858:
848:
838:
809:Lie algebras
626:
614:
602:
590:
578:
574:
566:
554:
542:
313:
270:
257:
246:
235:
231:Cyclic group
109:
96:Free product
67:Group action
30:Group theory
25:Group theory
24:
3543:. pp.
3404:space group
3350:Glide plane
3303:Translation
3222:Translation
3171:Translation
2968:(e.g., for
2506:composition
2371:), and the
2321:Felix Klein
2212:, they are
2008:defined by
1859:converges.
1493:reflections
1402:mathematics
1323:Ălie Cartan
1169:Root system
973:Exceptional
516:Topological
355:alternating
3589:Lie groups
3583:Categories
3529:References
3408:involution
3269:See also:
2753:reflection
2253:by a pair
2247:a fortiori
2216:, so that
2214:Lie groups
2134:rigid body
1641:handedness
1414:isometries
1308:Sophus Lie
1301:Scientists
1174:Weyl group
895:Symplectic
855:Orthogonal
805:Lie groups
623:Symplectic
563:Orthogonal
520:Lie groups
427:Free group
152:continuous
91:Direct sum
3398:See also
3352:operation
2922:Subgroups
2890:⋉
2792:≅
2665:≅
2601:⋊
2550:) is the
2520:−
2445:↦
2394:↦
1984:→
1891:→
1800:∈
1659:) or SE(
1489:rotations
1468:) or ISO(
1185:Real form
1071:Euclidean
922:Classical
687:Conformal
575:Euclidean
182:nilpotent
3492:See also
3292:Identity
3211:Identity
3160:Identity
2985:lattices
2966:discrete
2749:rotation
2367:group T(
2337:distance
2301:of size
2290:of size
2218:calculus
1716:topology
1698:) is of
1672:opposite
1668:indirect
1649:subgroup
1566:Overview
1547:) and E(
1537:indirect
1357:Glossary
1051:Poincaré
682:Poincaré
527:Solenoid
399:Integers
389:Lattices
364:sporadic
359:Lie type
187:solvable
177:dihedral
162:additive
147:infinite
57:Subgroup
3468:In 3D:
2999:√
2716:), the
2712:Now SO(
2639:) by T(
2488:) is a
2286:a real
2265:, with
1702:2 in E(
1476:group.
1472:), for
1408:is the
1265:physics
1046:Lorentz
875:Unitary
677:Lorentz
599:Unitary
498:Lattice
438:PSL(2,
172:abelian
83:(Semi-)
3572:
3551:
3547:â164.
3111:-fold
3016:e.g.:
2761:origin
2757:mirror
2744:is 1.
2738:direct
2504:, the
2428:is an
2424:where
2282:, and
2188:(0) =
2105:(1) =
2097:(0) =
2044:) = E(
1645:chiral
1625:â 1)/2
1592:+ 1)/2
1579:for E(
1560:groups
1558:These
1533:direct
1491:, and
1041:Circle
532:Circle
463:SL(2,
352:cyclic
316:-group
167:cyclic
142:finite
137:simple
121:kernel
3113:helix
2946:and I
2726:index
2724:) of
2635:of E(
2554:of O(
2492:of E(
2472:with
2341:angle
2089:in E(
2021:) = (
1700:index
1686:of E(
1684:coset
1670:, or
1651:of E(
1583:) is
1416:of a
1410:group
1116:Index
716:Sp(â)
713:SU(â)
126:image
3570:ISBN
3549:ISBN
3333:Yes
3320:Yes
3309:Yes
3298:Yes
3239:Yes
3228:Yes
3217:Yes
3177:Yes
3166:Yes
2970:1 â€
2294:; or
2236:for
2147:of
2101:and
2052:and
1404:, a
1066:Loop
807:and
710:O(â)
699:Loop
518:and
3545:136
3382:No
3371:No
3358:No
3344:No
3263:No
3250:No
3188:No
3105:â„ 1
2930:):
2767:).
2643:):
2323:'s
2306:+ 1
2269:an
2125:).
2056:of
2037:).
2029:))(
1919:of
1821:of
1706:).
1643:of
1627:to
1606:= 3
1599:= 2
1535:or
1495:of
1400:In
897:Sp(
887:SU(
867:SO(
847:SL(
837:GL(
625:Sp(
613:SU(
589:SO(
553:SL(
541:GL(
3585::
3564:.
3410:.
3406:,
3402:,
3395:.
3140::
2974:â€
2877:SO
2779:SO
2484:T(
2479:Ab
2477:=
2375:O(
2360:.
2274:Ă
2259:,
1631:.
1487:,
877:U(
857:O(
601:U(
577:E(
565:O(
23:â
3557:.
3379:3
3368:6
3355:5
3341:3
3330:6
3317:5
3306:3
3295:0
3260:3
3247:2
3236:3
3225:2
3214:0
3185:1
3174:1
3163:0
3117:k
3109:k
3103:k
3098:k
3072:m
3070:â
3068:n
3064:m
3057:m
3055:â
3053:n
3049:m
3044:)
3042:n
3037:)
3035:n
3030:)
3023:)
3001:2
2991:.
2981:m
2976:n
2972:m
2955:h
2949:h
2943:h
2937:.
2928:n
2907:.
2904:)
2901:n
2898:(
2894:T
2887:)
2884:n
2881:(
2873:=
2870:)
2867:n
2864:(
2859:+
2854:E
2832:)
2829:n
2826:(
2822:T
2817:/
2813:)
2810:n
2807:(
2802:+
2797:E
2789:)
2786:n
2783:(
2742:A
2734:n
2730:n
2722:n
2714:n
2698:)
2695:n
2692:(
2688:T
2683:/
2679:)
2676:n
2673:(
2669:E
2662:)
2659:n
2656:(
2652:O
2641:n
2637:n
2629:n
2615:)
2612:n
2609:(
2605:O
2598:)
2595:n
2592:(
2588:T
2584:=
2581:)
2578:n
2575:(
2571:E
2560:n
2556:n
2548:n
2531:u
2528:t
2523:1
2516:u
2502:u
2498:t
2494:n
2486:n
2475:c
2460:,
2457:c
2454:+
2451:x
2448:A
2442:x
2426:A
2412:)
2409:b
2406:+
2403:x
2400:(
2397:A
2391:x
2381:n
2377:n
2369:n
2312:.
2304:n
2292:n
2284:b
2276:n
2272:n
2267:A
2263:)
2261:b
2257:A
2255:(
2238:n
2230:n
2198:t
2196:(
2194:f
2190:I
2186:f
2182:f
2178:t
2162:3
2157:E
2145:I
2138:f
2123:n
2119:n
2115:n
2111:n
2107:B
2103:f
2099:A
2095:f
2091:n
2087:f
2071:n
2066:E
2054:B
2050:A
2046:n
2042:n
2035:n
2031:p
2027:t
2025:(
2023:f
2019:t
2017:(
2014:p
2010:f
1994:n
1989:E
1981:]
1978:1
1975:,
1972:0
1969:[
1966::
1961:p
1957:f
1934:n
1929:E
1917:p
1903:)
1900:n
1897:(
1894:E
1888:]
1885:1
1882:,
1879:0
1876:[
1873::
1870:f
1856:i
1852:p
1836:n
1831:E
1819:p
1804:N
1797:i
1789:(
1775:n
1770:E
1757:i
1753:f
1749:n
1733:n
1728:E
1704:n
1696:n
1692:n
1688:n
1680:R
1676:R
1661:n
1657:n
1653:n
1623:n
1621:(
1619:n
1610:n
1604:n
1597:n
1590:n
1588:(
1586:n
1581:n
1549:n
1545:n
1510:n
1505:E
1481:n
1470:n
1466:n
1462:n
1434:n
1429:E
1389:e
1382:t
1375:v
1015:8
1013:E
1007:7
1005:E
999:6
997:E
991:4
989:F
983:2
981:G
963:n
960:D
953:n
950:C
943:n
940:B
933:n
930:A
901:)
899:n
891:)
889:n
881:)
879:n
871:)
869:n
861:)
859:n
851:)
849:n
841:)
839:n
787:e
780:t
773:v
669:8
667:E
661:7
659:E
653:6
651:E
645:4
643:F
637:2
635:G
629:)
627:n
617:)
615:n
605:)
603:n
593:)
591:n
581:)
579:n
569:)
567:n
557:)
555:n
545:)
543:n
485:)
472:Z
460:)
447:Z
423:)
410:Z
401:(
314:p
279:Q
271:n
268:D
258:n
255:A
247:n
244:S
236:n
233:Z
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