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Group Theory

Expanded Small Groups Database and Database Search Commands

The database of small groups included in the GroupTheory package has been expanded to include all groups of order less than $512$ . Previously, only groups up to order $200$ were included.

Searching of the small groups database and the database of transitive groups are implemented in the new SearchSmallGroups and SearchTransitiveGroups commands, respectively.

>	$with (GroupTheory) &colon;$

Find the non-Abelian simple groups in the database; a sequence of small group IDs is returned.

>	$SearchSmallGroups (simple, abelian = false)$

$[60, 5], [168, 42], [360, 118], [504, 156]$

(1)

Find groups in the database of order less than $30$ with Sylow $2$ -subgroup of order $4$ , and an unique Sylow $3$ -subgroup, and output them as finitely presented groups.

>	$SearchSmallGroups (order < 30, {sylow}_{2} = 4, {nsylow}_{3} = 1, form = fpgroup)$

$⟨a1, a2, a3 ∣ {a2}^{2}, {a1}^{2} {a2}^{-1}, {a3}^{3}, {a2}^{-1} {a1}^{-1} a2 a1, {a3}^{-1} {a2}^{-1} a3 a2, {a3}^{-1} {a1}^{-1} a3 a1 {a3}^{-1}⟩, ⟨a1 ∣ {a1}^{12}⟩, ⟨a1, a2, a3 ∣ {a1}^{2}, {a2}^{2}, {a3}^{3}, {a2}^{-1} {a1}^{-1} a2 a1, {a3}^{-1} {a2}^{-1} a3 a2, {a3}^{-1} {a1}^{-1} a3 a1 {a3}^{-1}⟩, ⟨a1, a2, a3 ∣ {a1}^{2}, {a2}^{2}, {a3}^{3}, {a1}^{-1} {a2}^{-1} a1 a2, {a1}^{-1} {a3}^{-1} a1 a3, {a2}^{-1} {a1}^{-1} a2 a1, {a2}^{-1} {a3}^{-1} a2 a3, {a3}^{-1} {a1}^{-1} a3 a1, {a3}^{-1} {a2}^{-1} a3 a2⟩$

(2)

Find how many non-nilpotent groups in the database have derived subgroup isomorphic to the quaternion group of order $8$ .

>	$SearchSmallGroups (derivedsubgroup = [8, 4], nilpotent = false, form = count)$

$101$

(3)

Output the transitive group IDs of the regular permutation groups of degree $6$ .

>	$SearchTransitiveGroups (degree = 6, isregular)$

$[6, 1], [6, 2]$

(4)

>	$G := TransitiveGroup (6, 2)$

$G := ⟨(1, 3, 5) (2, 4, 6), (1, 4) (2, 3) (5, 6)⟩$

(5)

>	$Degree (G)$

$6$

(6)

>	$IsRegular (G)$

$true$

(7)

New Cayley Graph Visualization

A new command for computing and visualizing Cayley graphs of small groups is included in the GroupTheory package.

>	$with (GraphTheory) &colon;$

>	$C := CayleyGraph (ElementaryGroup (2, 3))$

$C := Graph 1: a directed unweighted graph with 8 vertices and 24 arc(s)$

(8)

>	$DrawGraph (C,' style' =' spring')$

>	$DrawGraph (CayleyGraph (Symm (4)),' style' =' spring')$

>	$DrawGraph (CayleyGraph (Alt (5)),' style' =' spring')$

>	$DrawGraph (CayleyGraph (PSL (2, 5)),' style' =' spring')$

>	$seq (DrawGraph (CayleyGraph (SmallGroup (12, i)),' style' =' spring'), i = 1 .. NumGroups (12))$

$PLOT (...), PLOT (...), PLOT (...), PLOT (...), PLOT (...)$

(9)

>	$Explore (DrawGraph (CayleyGraph (DihedralGroup (n)),' style' =' spring'),' parameters' = [' n' = 3 .. 20],' placement' =' right')$

$n$

>	$Explore (DrawGraph (CayleyGraph (MetacyclicGroup (m, n, k)),' style' =' spring'),' parameters' = [' m' = 3 .. 20,' n' = 2 .. 20,' k' = 2 .. 20],' initialvalues' = [' m' = 7,' n' = 6,' k' = 4],' placement' =' right')$

$m$

$n$

$k$

Other New Commands

The following additional commands are new in this release: ComplexProduct, ElementOrder, Exponent, FreeGroup, IsCyclic,

A complex in a group is just a subset of a group. The ComplexProduct command computes the product of two complexes $A$ and $B$ in a group $G$ , which is defined to be the set of products of the form $a &period; b$ , for $a$ in $A$ and $b$ in $B$ .

>	$G := Alt (6)$

$G := A_{6}$

(10)

>	$S := SylowSubgroup (5, G) &colon;$

>	$g := RandomElement (G) &colon;$

>	$g := RandomElement (G) &colon;$

>	$A := Elements (LeftCoset (g, S)) &colon;$

>	$g := RandomElement (G) &colon;$

>	$B := Elements (LeftCoset (g, S)) &colon;$

>	$ComplexProduct (A, B, G)$

$\{(2, 4, 6), (1, 2, 3, 6, 4), (1, 3, 4, 2, 6), (1, 4, 2, 5, 3), (1, 4, 3, 6, 2), (1, 5, 3, 2, 4), (1, 5, 4, 6, 3), (1, 5, 6, 4, 2), (1, 6, 4, 5, 3), (2, 3, 4, 5, 6), (2, 3, 6, 5, 4), (2, 6, 4, 3, 5), (1, 2) (4, 5), (1, 2) (3, 4, 6, 5), (1, 4) (5, 6), (1, 5) (3, 4), (1, 6) (2, 5, 4, 3), (1, 3, 6) (2, 4, 5), (1, 4, 5) (2, 6, 3), (1, 4, 6) (2, 3, 5), (1, 2, 4, 3) (5, 6), (1, 3, 2, 5) (4, 6), (1, 3, 5, 4) (2, 6), (1, 6, 3, 4) (2, 5), (1, 6, 3, 5) (2, 4)\}$

(11)

>	$g := RandomElement (G) &colon;$

>	$B := Elements (RightCoset (S, g)) &colon;$

>	$ComplexProduct (B, A, G)$

$\{(1, 2, 4), (1, 4, 3), (2, 5, 4), (3, 4, 6), (4, 5, 6), (1, 3, 4, 5, 2), (1, 3, 6, 4, 2), (1, 3, 6, 5, 4), (1, 4, 6, 5, 2), (2, 4, 3, 6, 5), (1, 4) (2, 5, 6, 3), (1, 5) (2, 6, 4, 3), (1, 6) (2, 4, 5, 3), (1, 3, 5) (2, 4, 6), (1, 4, 5) (2, 3, 6), (1, 6, 2) (3, 5, 4), (1, 6, 4) (2, 3, 5), (1, 6, 5) (2, 3, 4), (1, 2, 5, 3) (4, 6), (1, 2, 5, 6) (3, 4), (1, 2, 6, 3) (4, 5), (1, 4, 2, 6) (3, 5), (1, 5, 3, 4) (2, 6), (1, 5, 4, 6) (2, 3), (1, 5, 6, 3) (2, 4)\}$

(12)

The new ElementOrder command computes the order of an element of a finite group, represented as either a permutation group or a Cayley table group.

>	$G := Symm (4) &colon;$

>	$p := Perm ([[1, 3], [2, 4]])$

$p := (1, 3) (2, 4)$

(13)

>	$ElementOrder (p, G) = PermOrder (p)$

$2 = 2$

(14)

>	$C := CayleyTableGroup (DihedralGroup (4))$

$C := < a Cayley table group with 8 elements >$

(15)

>	$ElementOrder (5, C)$

$2$

(16)

The Exponent command computes the exponent of a finite group represented as either a permutation group or a Cayley table group.

>	$G := ElementaryGroup (3, 4)$

$G := C_{3}^{4}$

(17)

>	$Exponent (G)$

$3$

(18)

>	$Exponent (DihedralGroup (6))$

$6$

(19)

>	$Exponent (Group (Perm ([[1, 2], [3, 4]]), Perm ([[1, 3], [2, 4]])))$

$2$

(20)

To construct a free group as a finitely presented group, use the new FreeGroup command.

>	$F := FreeGroup (3)$

$F := ⟨_x1,_x2,_x3 ∣⟩$

(21)

>	$type (F,' FPGroup')$

$true$

(22)

>	$IsAbelian (F)$

$false$

(23)

The IsCyclic command attempts to determine whether a group is cyclic.

>	$IsCyclic (Group (Perm ([[1, 2], [3, 4]]), Perm ([[1, 3], [2, 4]])))$

$false$

(24)

>	$G := DirectProduct (CyclicGroup (2), CyclicGroup (3))$

$G := C_{2} \times C_{3}$

(25)

>	$IsCyclic (G)$

$true$

(26)

>	$G := DirectProduct (CyclicGroup (2), CyclicGroup (4))$

$G := C_{2} \times C_{4}$

(27)

>	$IsCyclic (G)$

$false$

(28)

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