GroupTheory/IsCPGroup

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GroupTheory

IsCPGroup

determine whether a group is a (CP)-group

IsCP1Group

determine whether a group is a (CP1)-group

	Calling Sequence
	IsCPGroup( G ) IsCP1Group( G )

Parameters

a group

Description

•

A group $G$ is a (CP)-group if each of its elements has prime-power order, where the prime may depend upon the element. A group of prime-power order is a (CP)-group, but there are (CP)-groups, such as $S_{3}$ , whose order is divisible by more than one prime. This is equivalent to the condition that the centralizer of each non-trivial element is a $p$ -group, for some prime $p$ depending upon the element. It is also equivalent for finite groups to the Gruenberg-Kegel graph of the group being totally disconnected.

•

A group $G$ is a (CP1)-group if each of its non-trivial elements has prime order where, again, the prime may depend upon the element. Equivalently, a group is a (CP1)-group if the centralizer of each non-trivial element contains only elements of order dividing $p$ , for a prime $p$ depending upon the element. The symmetric group $S_{3}$ is again an example of a (CP1)-group not of prime exponent.

•	It is a consequence of these definitions that every (CP1)-group is a (CP)-group. Both (CP)-groups and (CP1)-groups are (CN)-groups.

•	The IsCPGroup( G ) command returns true if the group G is a (CP)-group and returns false otherwise.

•	The IsCP1Group( G ) command returns true if the group G is a (CP1)-group and returns false otherwise.

Examples

>	$with (GroupTheory) &colon;$

The symmetric group $S_{3}$ furnishes an example of a (CP)-group that is not a group of prime power order, and a (CP1)-group that is not of prime exponent.

>	$IsCPGroup (Symm (3))$

$true$

(1)

>	$IsPGroup (Symm (3))$

$false$

(2)

>	$IsCP1Group (Symm (3))$

$true$

(3)

>	$Exponent (Symm (3))$

$6$

(4)

The symmetric group $S_{4}$ is also a (CP)-group, but symmetric groups of larger degree are not. However, $S_{4}$ is not a (CP1)-group.

>	$IsCPGroup (Symm (4))$

$true$

(5)

>	$IsCP1Group (Symm (4))$

$false$

(6)

>	$orseq (IsCPGroup (Symm (n)), n = 5 .. 10)$

$false$

(7)

A cyclic group is a (CP)-group if, and only if, it has prime power order.

>	$seq (n = IsCPGroup (CyclicGroup (n)), n = 1 .. 20)$

$1 = true, 2 = true, 3 = true, 4 = true, 5 = true, 6 = false, 7 = true, 8 = true, 9 = true, 10 = false, 11 = true, 12 = false, 13 = true, 14 = false, 15 = false, 16 = true, 17 = true, 18 = false, 19 = true, 20 = false$

(8)

And cyclic groups are (CP1)-groups precisely when the order is a prime.

>	$seq (n = IsCP1Group (CyclicGroup (n)), n = 1 .. 20)$

$1 = true, 2 = true, 3 = true, 4 = false, 5 = true, 6 = false, 7 = true, 8 = false, 9 = false, 10 = false, 11 = true, 12 = false, 13 = true, 14 = false, 15 = false, 16 = false, 17 = true, 18 = false, 19 = true, 20 = false$

(9)

Dihedral groups are (CP)-groups just when the degree is a prime power.

>	$seq (n = IsCPGroup (DihedralGroup (n)), n = 1 .. 20)$

(10)

And, dihedral groups of prime degree are the only ones that are (CP1)-groups.

>	$seq (n = IsCP1Group (DihedralGroup (n)), n = 1 .. 20)$

(11)

Groups of prime exponent are (CP1)-groups.

>	$G ≔ SmallGroup (81, 12) &colon;$

>	$Exponent (G)$

$3$

(12)

>	$IsCP1Group (G)$

$true$

(13)

>	$G ≔ SmallGroup (98, 4)$

$G ≔ ⟨(1, 2) (3, 8) (4, 7) (5, 10) (6, 9) (11, 22) (12, 24) (13, 23) (14, 19) (15, 21) (16, 20) (17, 26) (18, 25) (27, 44) (28, 46) (29, 45) (30, 48) (31, 47) (32, 39) (33, 41) (34, 40) (35, 43) (36, 42) (37, 50) (38, 49) (51, 70) (52, 69) (53, 72) (54, 71) (55, 74) (56, 73) (57, 64) (58, 63) (59, 66) (60, 65) (61, 68) (62, 67) (75, 88) (76, 87) (77, 90) (78, 89) (79, 84) (80, 83) (81, 86) (82, 85) (91, 98) (92, 97) (93, 96) (94, 95), (1, 3, 11, 27, 32, 14, 4) (2, 7, 19, 39, 44, 22, 8) (5, 12, 28, 51, 57, 33, 15) (6, 13, 29, 52, 58, 34, 16) (9, 20, 40, 63, 69, 45, 23) (10, 21, 41, 64, 70, 46, 24) (17, 30, 53, 75, 79, 59, 35) (18, 31, 54, 76, 80, 60, 36) (25, 42, 65, 83, 87, 71, 47) (26, 43, 66, 84, 88, 72, 48) (37, 55, 77, 91, 93, 81, 61) (38, 56, 78, 92, 94, 82, 62) (49, 67, 85, 95, 97, 89, 73) (50, 68, 86, 96, 98, 90, 74), (1, 5, 17, 37, 38, 18, 6) (2, 9, 25, 49, 50, 26, 10) (3, 12, 30, 55, 56, 31, 13) (4, 15, 35, 61, 62, 36, 16) (7, 20, 42, 67, 68, 43, 21) (8, 23, 47, 73, 74, 48, 24) (11, 28, 53, 77, 78, 54, 29) (14, 33, 59, 81, 82, 60, 34) (19, 40, 65, 85, 86, 66, 41) (22, 45, 71, 89, 90, 72, 46) (27, 51, 75, 91, 92, 76, 52) (32, 57, 79, 93, 94, 80, 58) (39, 63, 83, 95, 96, 84, 64) (44, 69, 87, 97, 98, 88, 70)⟩$

(14)

>	$IsCPGroup (G)$

$true$

(15)

>	$IsCP1Group (G)$

$true$

(16)

>	$G ≔ SmallGroup (72, 41)$

$G ≔ ⟨(1, 2, 6, 3) (4, 12, 5, 11) (7, 33, 30, 34) (8, 37, 29, 38) (9, 41, 32, 42) (10, 43, 31, 44) (13, 53, 18, 39) (14, 54, 17, 36) (15, 55, 20, 35) (16, 56, 19, 40) (21, 57, 26, 58) (22, 61, 25, 62) (23, 65, 28, 66) (24, 67, 27, 68) (45, 60, 50, 71) (46, 63, 49, 70) (47, 64, 52, 69) (48, 59, 51, 72), (1, 4, 6, 5) (2, 11, 3, 12) (7, 23, 30, 28) (8, 24, 29, 27) (9, 22, 32, 25) (10, 21, 31, 26) (13, 47, 18, 52) (14, 48, 17, 51) (15, 46, 20, 49) (16, 45, 19, 50) (33, 66, 34, 65) (35, 63, 55, 70) (36, 59, 54, 72) (37, 68, 38, 67) (39, 64, 53, 69) (40, 60, 56, 71) (41, 62, 42, 61) (43, 58, 44, 57), (1, 6) (2, 3) (4, 5) (7, 30) (8, 29) (9, 32) (10, 31) (11, 12) (13, 18) (14, 17) (15, 20) (16, 19) (21, 26) (22, 25) (23, 28) (24, 27) (33, 34) (35, 55) (36, 54) (37, 38) (39, 53) (40, 56) (41, 42) (43, 44) (45, 50) (46, 49) (47, 52) (48, 51) (57, 58) (59, 72) (60, 71) (61, 62) (63, 70) (64, 69) (65, 66) (67, 68), (1, 7, 8) (2, 13, 14) (3, 17, 18) (4, 21, 22) (5, 25, 26) (6, 29, 30) (9, 35, 39) (10, 36, 40) (11, 45, 46) (12, 49, 50) (15, 43, 37) (16, 33, 41) (19, 42, 34) (20, 38, 44) (23, 59, 63) (24, 60, 64) (27, 69, 71) (28, 70, 72) (31, 56, 54) (32, 53, 55) (47, 66, 58) (48, 62, 68) (51, 67, 61) (52, 57, 65), (1, 9, 10) (2, 15, 16) (3, 19, 20) (4, 23, 24) (5, 27, 28) (6, 31, 32) (7, 35, 36) (8, 39, 40) (11, 47, 48) (12, 51, 52) (13, 43, 33) (14, 37, 41) (17, 42, 38) (18, 34, 44) (21, 59, 60) (22, 63, 64) (25, 69, 70) (26, 71, 72) (29, 56, 53) (30, 54, 55) (45, 66, 62) (46, 58, 68) (49, 67, 57) (50, 61, 65)⟩$

(17)

>	$IsCPGroup (G)$

$true$

(18)

>	$IsCP1Group (G)$

$false$

(19)

>	$G ≔ SmallGroup (192, 182)$

$G ≔ &lang;a permutation group on 192 letters with 7 generators&rang;$

(20)

>	$IsCPGroup (G)$

$false$

(21)

>	$Index (PCore (2, G), G)$

$6$

(22)

The group ${}^{2}B_{2} (2)$ is the Frobenius group of order $20$ and is a (CP)-group.

>	$IsCPGroup (Suzuki2B2 (2))$

$true$

(23)

Other Frobenius groups provide important examples of (CP)-groups.

>	$IsCPGroup (FrobeniusGroup (72, 2))$

$true$

(24)

These are all the simple (CP)-groups (M. Suzuki).

>	$IsCPGroup (Alt (5))$

$true$

(25)

>	$IsCPGroup (Suzuki2B2 (8))$

$true$

(26)

>	$IsCPGroup (Suzuki2B2 (32))$

$true$

(27)

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

$true$

(28)

>	$IsCPGroup (PSL (2, 8))$

$true$

(29)

>	$IsCPGroup (PSL (2, 9))$

$true$

(30)

>	$IsCPGroup (PSL (2, 17))$

$true$

(31)

>	$IsCPGroup (PSL (3, 4))$

$true$

(32)

>	$GraphTheory :- DrawGraph (GruenbergKegelGraph (PSL (3, 4)))$

The only simple (in fact, the only insoluble) (CP1)-group is the alternating group $A_{5}$ .

>	$IsCP1Group (Alt (5))$

$true$

(33)

An example of an infinite (CP)-group.

>	$G ≔ QuasicyclicGroup (17)$

$G ≔ ℤ_{17^{\infty}}$

(34)

>	$IsCPGroup (G)$

$true$

(35)

>	$GroupOrder (G)$

$\infty$

(36)

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