IdentifySmallGroup - Maple Help

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GroupTheory

IdentifySmallGroup

find where a group is in the small groups database

	Calling Sequence
	IdentifySmallGroup(G, opts)

Parameters

G	-	a group
opts	-	(optional) equations of the form keyword = value, listed below

Options

•	assign = name

If given the option assign = x, where x is any name, IdentifySmallGroup will assign the isomorphism mapping $G$ to $H$ to the name x. This isomorphism can be used in the same way as the isomorphisms assigned by AreIsomorphic.

If x already has a value, then it needs to be protected from evaluation using quotation marks.

•	form = fpgroup or form = permgroup

This option can be used together with the assign option, explained above, in order to specify the form of the group $H$ that is the codomain of the isomorphism to be assigned to the name specified in the assign option.

Specifying form = fpgroup results in the codomain being a finitely presented group. Specifying form = permgroup (the default) results in the codomain being a permutation group. You can equivalently specify the string forms of these values, as form = "fpgroup" or form = "permgroup".

If no assign option is specified, then the form option is ignored.

Description

•	The command IdentifySmallGroup finds if a group $H$ isomorphic to $G$ occurs in the small groups database. (Currently, that means that the order of the group is at most 511.) If so, it returns the numbers under which $H$ occurs in the database.

•	The value returned is a sequence of two numbers such that calling SmallGroup with those two numbers as arguments returns the group $H$ . The first number is the order of $G$ .

Examples

>	$with (GroupTheory) &colon;$

We identify the three-dimensional projective special linear group over the field of two elements.

>	$IdentifySmallGroup (PSL (3, 2))$

$168, 42$

(1)

>	$IdentifySmallGroup (PSL (2, 7))$

$168, 42$

(2)

We see that both groups are isomorphic (because they are both isomorphic to $SmallGroup (168, 42)$ ). Now construct a group using the $SmallGroup$ command, then create a Cayley table group that is isomorphic to it, and test that it is still recognized as the same group.

>	$g1 ≔ SmallGroup (96, 7)$

$g1 ≔ ⟨(1, 2) (3, 21) (4, 30) (5, 15) (6, 14) (7, 17) (8, 16) (9, 18) (10, 20) (11, 19) (12, 23) (13, 27) (22, 51) (24, 47) (25, 46) (26, 52) (28, 66) (29, 65) (31, 45) (32, 48) (33, 80) (34, 79) (35, 58) (36, 57) (37, 56) (38, 55) (39, 62) (40, 61) (41, 60) (42, 59) (43, 64) (44, 63) (49, 70) (50, 69) (53, 78) (54, 77) (67, 94) (68, 93) (71, 90) (72, 89) (73, 88) (74, 87) (75, 96) (76, 95) (81, 86) (82, 85) (83, 92) (84, 91), (1, 3, 22, 32, 8, 26, 5, 23, 9, 27, 6, 24, 7, 25, 31, 4) (2, 12, 45, 52, 17, 48, 14, 21, 18, 30, 15, 46, 16, 47, 51, 13) (10, 28, 67, 83, 41, 75, 35, 69, 43, 77, 37, 71, 39, 73, 81, 33) (11, 29, 68, 84, 42, 76, 36, 70, 44, 78, 38, 72, 40, 74, 82, 34) (19, 49, 85, 95, 61, 91, 55, 65, 63, 79, 57, 87, 59, 89, 93, 53) (20, 50, 86, 96, 62, 92, 56, 66, 64, 80, 58, 88, 60, 90, 94, 54), (1, 5, 7, 22, 9, 31, 8, 6) (2, 14, 16, 45, 18, 51, 17, 15) (3, 23, 25, 32, 27, 4, 26, 24) (10, 35, 39, 67, 43, 81, 41, 37) (11, 36, 40, 68, 44, 82, 42, 38) (12, 21, 47, 52, 30, 13, 48, 46) (19, 55, 59, 85, 63, 93, 61, 57) (20, 56, 60, 86, 64, 94, 62, 58) (28, 69, 73, 83, 77, 33, 75, 71) (29, 70, 74, 84, 78, 34, 76, 72) (49, 65, 89, 95, 79, 53, 91, 87) (50, 66, 90, 96, 80, 54, 92, 88), (1, 7, 9, 8) (2, 16, 18, 17) (3, 25, 27, 26) (4, 24, 23, 32) (5, 22, 31, 6) (10, 39, 43, 41) (11, 40, 44, 42) (12, 47, 30, 48) (13, 46, 21, 52) (14, 45, 51, 15) (19, 59, 63, 61) (20, 60, 64, 62) (28, 73, 77, 75) (29, 74, 78, 76) (33, 71, 69, 83) (34, 72, 70, 84) (35, 67, 81, 37) (36, 68, 82, 38) (49, 89, 79, 91) (50, 90, 80, 92) (53, 87, 65, 95) (54, 88, 66, 96) (55, 85, 93, 57) (56, 86, 94, 58), (1, 9) (2, 18) (3, 27) (4, 23) (5, 31) (6, 22) (7, 8) (10, 43) (11, 44) (12, 30) (13, 21) (14, 51) (15, 45) (16, 17) (19, 63) (20, 64) (24, 32) (25, 26) (28, 77) (29, 78) (33, 69) (34, 70) (35, 81) (36, 82) (37, 67) (38, 68) (39, 41) (40, 42) (46, 52) (47, 48) (49, 79) (50, 80) (53, 65) (54, 66) (55, 93) (56, 94) (57, 85) (58, 86) (59, 61) (60, 62) (71, 83) (72, 84) (73, 75) (74, 76) (87, 95) (88, 96) (89, 91) (90, 92), (1, 10, 11) (2, 19, 20) (3, 28, 29) (4, 33, 34) (5, 35, 36) (6, 37, 38) (7, 39, 40) (8, 41, 42) (9, 43, 44) (12, 49, 50) (13, 53, 54) (14, 55, 56) (15, 57, 58) (16, 59, 60) (17, 61, 62) (18, 63, 64) (21, 65, 66) (22, 67, 68) (23, 69, 70) (24, 71, 72) (25, 73, 74) (26, 75, 76) (27, 77, 78) (30, 79, 80) (31, 81, 82) (32, 83, 84) (45, 85, 86) (46, 87, 88) (47, 89, 90) (48, 91, 92) (51, 93, 94) (52, 95, 96)⟩$

(3)

>	$g2 ≔ CayleyTableGroup (g1)$

$g2 ≔ < a Cayley table group with 96 elements >$

(4)

>	$IdentifySmallGroup (g2,'assign = iso')$

$96, 7$

(5)

>	$Domain (iso)$

$< a Cayley table group with 96 elements >$

(6)

>	$Codomain (iso)$

$⟨(1, 2) (3, 21) (4, 30) (5, 15) (6, 14) (7, 17) (8, 16) (9, 18) (10, 20) (11, 19) (12, 23) (13, 27) (22, 51) (24, 47) (25, 46) (26, 52) (28, 66) (29, 65) (31, 45) (32, 48) (33, 80) (34, 79) (35, 58) (36, 57) (37, 56) (38, 55) (39, 62) (40, 61) (41, 60) (42, 59) (43, 64) (44, 63) (49, 70) (50, 69) (53, 78) (54, 77) (67, 94) (68, 93) (71, 90) (72, 89) (73, 88) (74, 87) (75, 96) (76, 95) (81, 86) (82, 85) (83, 92) (84, 91), (1, 3, 22, 32, 8, 26, 5, 23, 9, 27, 6, 24, 7, 25, 31, 4) (2, 12, 45, 52, 17, 48, 14, 21, 18, 30, 15, 46, 16, 47, 51, 13) (10, 28, 67, 83, 41, 75, 35, 69, 43, 77, 37, 71, 39, 73, 81, 33) (11, 29, 68, 84, 42, 76, 36, 70, 44, 78, 38, 72, 40, 74, 82, 34) (19, 49, 85, 95, 61, 91, 55, 65, 63, 79, 57, 87, 59, 89, 93, 53) (20, 50, 86, 96, 62, 92, 56, 66, 64, 80, 58, 88, 60, 90, 94, 54), (1, 5, 7, 22, 9, 31, 8, 6) (2, 14, 16, 45, 18, 51, 17, 15) (3, 23, 25, 32, 27, 4, 26, 24) (10, 35, 39, 67, 43, 81, 41, 37) (11, 36, 40, 68, 44, 82, 42, 38) (12, 21, 47, 52, 30, 13, 48, 46) (19, 55, 59, 85, 63, 93, 61, 57) (20, 56, 60, 86, 64, 94, 62, 58) (28, 69, 73, 83, 77, 33, 75, 71) (29, 70, 74, 84, 78, 34, 76, 72) (49, 65, 89, 95, 79, 53, 91, 87) (50, 66, 90, 96, 80, 54, 92, 88), (1, 7, 9, 8) (2, 16, 18, 17) (3, 25, 27, 26) (4, 24, 23, 32) (5, 22, 31, 6) (10, 39, 43, 41) (11, 40, 44, 42) (12, 47, 30, 48) (13, 46, 21, 52) (14, 45, 51, 15) (19, 59, 63, 61) (20, 60, 64, 62) (28, 73, 77, 75) (29, 74, 78, 76) (33, 71, 69, 83) (34, 72, 70, 84) (35, 67, 81, 37) (36, 68, 82, 38) (49, 89, 79, 91) (50, 90, 80, 92) (53, 87, 65, 95) (54, 88, 66, 96) (55, 85, 93, 57) (56, 86, 94, 58), (1, 9) (2, 18) (3, 27) (4, 23) (5, 31) (6, 22) (7, 8) (10, 43) (11, 44) (12, 30) (13, 21) (14, 51) (15, 45) (16, 17) (19, 63) (20, 64) (24, 32) (25, 26) (28, 77) (29, 78) (33, 69) (34, 70) (35, 81) (36, 82) (37, 67) (38, 68) (39, 41) (40, 42) (46, 52) (47, 48) (49, 79) (50, 80) (53, 65) (54, 66) (55, 93) (56, 94) (57, 85) (58, 86) (59, 61) (60, 62) (71, 83) (72, 84) (73, 75) (74, 76) (87, 95) (88, 96) (89, 91) (90, 92), (1, 10, 11) (2, 19, 20) (3, 28, 29) (4, 33, 34) (5, 35, 36) (6, 37, 38) (7, 39, 40) (8, 41, 42) (9, 43, 44) (12, 49, 50) (13, 53, 54) (14, 55, 56) (15, 57, 58) (16, 59, 60) (17, 61, 62) (18, 63, 64) (21, 65, 66) (22, 67, 68) (23, 69, 70) (24, 71, 72) (25, 73, 74) (26, 75, 76) (27, 77, 78) (30, 79, 80) (31, 81, 82) (32, 83, 84) (45, 85, 86) (46, 87, 88) (47, 89, 90) (48, 91, 92) (51, 93, 94) (52, 95, 96)⟩$

(7)

Using the infolevel facility, we can obtain some information about the progress of the command.

>	$infolevel [GroupTheory] ≔ 3$

${infolevel}_{GroupTheory} ≔ 3$

(8)

>	$g3 ≔ SmallGroup (128, 1607)$

$g3 ≔ ⟨(1, 2, 8, 16, 11, 18, 9, 3) (4, 24, 28, 19, 30, 12, 29, 25) (5, 31, 35, 20, 37, 13, 36, 32) (6, 14, 39, 56, 42, 58, 40, 21) (7, 15, 44, 60, 47, 62, 45, 22) (10, 17, 48, 63, 50, 64, 49, 23) (26, 73, 78, 65, 80, 51, 79, 75) (27, 74, 82, 66, 84, 52, 83, 76) (33, 85, 90, 67, 92, 53, 91, 87) (34, 86, 94, 68, 96, 54, 95, 88) (38, 55, 97, 71, 43, 59, 98, 69) (41, 57, 100, 112, 102, 113, 101, 70) (46, 61, 104, 115, 106, 116, 105, 72) (77, 108, 123, 117, 81, 107, 124, 118) (89, 110, 125, 119, 93, 109, 126, 120) (99, 111, 127, 122, 103, 114, 128, 121), (1, 4, 10, 5) (2, 12, 17, 13) (3, 19, 23, 20) (6, 26, 41, 33) (7, 27, 46, 34) (8, 28, 48, 35) (9, 29, 49, 36) (11, 30, 50, 37) (14, 51, 57, 53) (15, 52, 61, 54) (16, 25, 63, 32) (18, 24, 64, 31) (21, 65, 70, 67) (22, 66, 72, 68) (38, 77, 99, 89) (39, 78, 100, 90) (40, 79, 101, 91) (42, 80, 102, 92) (43, 81, 103, 93) (44, 82, 104, 94) (45, 83, 105, 95) (47, 84, 106, 96) (55, 107, 111, 109) (56, 75, 112, 87) (58, 73, 113, 85) (59, 108, 114, 110) (60, 76, 115, 88) (62, 74, 116, 86) (69, 117, 121, 119) (71, 118, 122, 120) (97, 123, 127, 125) (98, 124, 128, 126), (1, 6) (2, 14) (3, 21) (4, 26) (5, 33) (7, 43) (8, 39) (9, 40) (10, 41) (11, 42) (12, 51) (13, 53) (15, 59) (16, 56) (17, 57) (18, 58) (19, 65) (20, 67) (22, 71) (23, 70) (24, 73) (25, 75) (27, 81) (28, 78) (29, 79) (30, 80) (31, 85) (32, 87) (34, 93) (35, 90) (36, 91) (37, 92) (38, 47) (44, 98) (45, 97) (46, 103) (48, 100) (49, 101) (50, 102) (52, 108) (54, 110) (55, 62) (60, 69) (61, 114) (63, 112) (64, 113) (66, 118) (68, 120) (72, 122) (74, 107) (76, 117) (77, 84) (82, 124) (83, 123) (86, 109) (88, 119) (89, 96) (94, 126) (95, 125) (99, 106) (104, 128) (105, 127) (111, 116) (115, 121), (1, 7) (2, 15) (3, 22) (4, 27) (5, 34) (6, 38) (8, 44) (9, 45) (10, 46) (11, 47) (12, 52) (13, 54) (14, 55) (16, 60) (17, 61) (18, 62) (19, 66) (20, 68) (21, 69) (23, 72) (24, 74) (25, 76) (26, 77) (28, 82) (29, 83) (30, 84) (31, 86) (32, 88) (33, 89) (35, 94) (36, 95) (37, 96) (39, 97) (40, 98) (41, 99) (42, 43) (48, 104) (49, 105) (50, 106) (51, 107) (53, 109) (56, 71) (57, 111) (58, 59) (63, 115) (64, 116) (65, 117) (67, 119) (70, 121) (73, 108) (75, 118) (78, 123) (79, 124) (80, 81) (85, 110) (87, 120) (90, 125) (91, 126) (92, 93) (100, 127) (101, 128) (102, 103) (112, 122) (113, 114), (1, 8, 11, 9) (2, 16, 18, 3) (4, 28, 30, 29) (5, 35, 37, 36) (6, 39, 42, 40) (7, 44, 47, 45) (10, 48, 50, 49) (12, 25, 24, 19) (13, 32, 31, 20) (14, 56, 58, 21) (15, 60, 62, 22) (17, 63, 64, 23) (26, 78, 80, 79) (27, 82, 84, 83) (33, 90, 92, 91) (34, 94, 96, 95) (38, 97, 43, 98) (41, 100, 102, 101) (46, 104, 106, 105) (51, 75, 73, 65) (52, 76, 74, 66) (53, 87, 85, 67) (54, 88, 86, 68) (55, 71, 59, 69) (57, 112, 113, 70) (61, 115, 116, 72) (77, 123, 81, 124) (89, 125, 93, 126) (99, 127, 103, 128) (107, 118, 108, 117) (109, 120, 110, 119) (111, 122, 114, 121), (1, 10) (2, 17) (3, 23) (4, 5) (6, 41) (7, 46) (8, 48) (9, 49) (11, 50) (12, 13) (14, 57) (15, 61) (16, 63) (18, 64) (19, 20) (21, 70) (22, 72) (24, 31) (25, 32) (26, 33) (27, 34) (28, 35) (29, 36) (30, 37) (38, 99) (39, 100) (40, 101) (42, 102) (43, 103) (44, 104) (45, 105) (47, 106) (51, 53) (52, 54) (55, 111) (56, 112) (58, 113) (59, 114) (60, 115) (62, 116) (65, 67) (66, 68) (69, 121) (71, 122) (73, 85) (74, 86) (75, 87) (76, 88) (77, 89) (78, 90) (79, 91) (80, 92) (81, 93) (82, 94) (83, 95) (84, 96) (97, 127) (98, 128) (107, 109) (108, 110) (117, 119) (118, 120) (123, 125) (124, 126), (1, 11) (2, 18) (3, 16) (4, 30) (5, 37) (6, 42) (7, 47) (8, 9) (10, 50) (12, 24) (13, 31) (14, 58) (15, 62) (17, 64) (19, 25) (20, 32) (21, 56) (22, 60) (23, 63) (26, 80) (27, 84) (28, 29) (33, 92) (34, 96) (35, 36) (38, 43) (39, 40) (41, 102) (44, 45) (46, 106) (48, 49) (51, 73) (52, 74) (53, 85) (54, 86) (55, 59) (57, 113) (61, 116) (65, 75) (66, 76) (67, 87) (68, 88) (69, 71) (70, 112) (72, 115) (77, 81) (78, 79) (82, 83) (89, 93) (90, 91) (94, 95) (97, 98) (99, 103) (100, 101) (104, 105) (107, 108) (109, 110) (111, 114) (117, 118) (119, 120) (121, 122) (123, 124) (125, 126) (127, 128)⟩$

(9)

>	$IdentifySmallGroup (g3)$

$128, 1607$

(10)

Compatibility

•	The GroupTheory[IdentifySmallGroup] command was introduced in Maple 17.

•	For more information on Maple 17 changes, see Updates in Maple 17.

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