The groups of order p 7 Eamonn O’Brien and Michael Vaughan-Lee The groups of order p 7 – p. 1
Groups of order p k for k = 1 , 2 , . . . , 6 p = 2 p = 3 p ≥ 5 p 1 1 1 p 2 2 2 2 p 3 5 5 5 p 4 14 15 15 p 5 51 67 u p 6 267 504 v u = 2 p + 61 + 2 gcd( p − 1 , 3) + gcd( p − 1 , 4) v = 3 p 2 +39 p +344+24 gcd( p − 1 , 3)+11 gcd( p − 1 , 4)+2 gcd( p − 1 , 5) The groups of order p 7 – p. 2
Order p 7 p = 2 p = 3 p = 5 2328 9310 34297 For p > 5 the number of groups of order p 7 is 3 p 5 + 12 p 4 + 44 p 3 + 170 p 2 + 707 p + 2455 +(4 p 2 + 44 p + 291) gcd( p − 1 , 3) +( p 2 + 19 p + 135) gcd( p − 1 , 4) +(3 p + 31) gcd( p − 1 , 5) +4 gcd( p − 1 , 7) + 5 gcd( p − 1 , 8) + gcd( p − 1 , 9) The groups of order p 7 – p. 3
Baker-Campbell-Hausdorff Formula e x . e y = e u where x + y − 1 2[ y, x ] + 1 12[ y, x, x ] − 1 12[ y, x, y ] + 1 u = 24[ y, x, x, y ] − 1 1 1 720[ y, x, x, x, x ] − 180[ y, x, x, x, y ] + 180[ y, x, x, y, y ] + 1 1 1 720[ y, x, y, y, y ] − 120[ y, x, x, [ y, x ]] − 360[ y, x, y, [ y, x ]] + . . The groups of order p 7 – p. 4
Baker-Campbell-Hausdorff Formula e x . e y = e u where x + y − 1 2[ y, x ] + 1 12[ y, x, x ] − 1 12[ y, x, y ] + 1 u = 24[ y, x, x, y ] − 1 1 1 720[ y, x, x, x, x ] − 180[ y, x, x, x, y ] + 180[ y, x, x, y, y ] + 1 1 1 720[ y, x, y, y, y ] − 120[ y, x, x, [ y, x ]] − 360[ y, x, y, [ y, x ]] + . . [e y , e x ] = e w where [ y, x ] + 1 2[ y, x, x ] + 1 w = 2[ y, x, y ] +1 6[ y, x, x, x ] + 1 4[ y, x, x, y ] + 1 6[ y, x, y, y ] + . . . The groups of order p 7 – p. 4
If L is a Lie algebra define a group operation ◦ on L by setting a ◦ b = a + b − 1 2[ b, a ] + 1 12[ b, a, a ] − 1 12[ b, a, b ] + . . . This works if L is a nilpotent Lie algebra over Q , or if L is a Lie ring of order p k and L is nilpotent of class at most p − 1 . The groups of order p 7 – p. 5
If G is a group under ◦ and if a, b ∈ G define G ◦ [ b, a, a ] − 1 1 1 a + b = a ◦ b ◦ [ b, a ] ◦ [ b, a, b ] G ◦ . . . 2 12 12 G [ b, a ] L = [ b, a ] G ◦ [ b, a, a ] − 1 G ◦ [ b, a, b ] − 1 G ◦ . . . 2 2 The groups of order p 7 – p. 6
If G is a group under ◦ and if a, b ∈ G define G ◦ [ b, a, a ] − 1 1 1 a + b = a ◦ b ◦ [ b, a ] ◦ [ b, a, b ] G ◦ . . . 2 12 12 G [ b, a ] L = [ b, a ] G ◦ [ b, a, a ] − 1 G ◦ [ b, a, b ] − 1 G ◦ . . . 2 2 We need G to be nilpotent, and we need unique extraction of roots. So this works if G is a nilpotent torsion free divisible group, or if G is a finite p -group of class at most p − 1 . The groups of order p 7 – p. 6
If G is a group under ◦ and if a, b ∈ G define G ◦ [ b, a, a ] − 1 1 1 a + b = a ◦ b ◦ [ b, a ] ◦ [ b, a, b ] G ◦ . . . 2 12 12 G [ b, a ] L = [ b, a ] G ◦ [ b, a, a ] − 1 G ◦ [ b, a, b ] − 1 G ◦ . . . 2 2 This gives the Mal’cev correspondence between nilpotent Lie algebras over Q and nilpotent torsion free divisible groups. It also gives the Lazard correspondence between nilpotent Lie rings of order p k and class at most p − 1 and finite groups of order p k and class at most p − 1 . The groups of order p 7 – p. 6
Classify groups of order p 7 for p > 5 by classifying nilpotent Lie rings of order p 7 . Use the Lie ring generation algorithm to classify the Lie rings. (Analogous to the p -group generation algorithm.) Then use the Baker-Campbell-Hausdorff formula to translate Lie ring presentations into group presentations. The groups of order p 7 – p. 7
Lower exponent- p -central series L 1 = L L 2 = pL + [ L, L ] L 3 = pL 2 + [ L 2 , L ] . . . L n +1 = pL n + [ L n , L ] a, b ba, pa, pb baa, bab, pba, p 2 a, p 2 b . . . The groups of order p 7 – p. 8
L has p -class c if L c +1 = { 0 } , L c � = { 0 } . Classify the nilpotent Lie rings of order p k according to p -class. If L has p -class c > 1 then we say that L is an immediate descendant of L/L c . To classify nilpotent Lie rings of order p k , first classify all nilpotent Lie rings of order p m for m < k . If L has order p m ( m < k ) find all immediate descendants of L of order p k . The groups of order p 7 – p. 9
The p -covering ring Let M be a nilpotent d -generator Lie ring of order p m The p -covering ring � M is the largest d -generator Lie ring with an ideal Z satisfying Z ≤ ζ ( � M ) pZ = { 0 } M/Z ∼ � = M The groups of order p 7 – p. 10
Immediate descendants If M has p -class c then every immediate descendant of M is of the form � M/T for some T < Z such that T + � M c +1 = Z If α is an automorphism of M then α lifts to an automorphism α ∗ of � M . M/S ∼ � = � M/T if and only if T = Sα ∗ for some α . The groups of order p 7 – p. 11
An example � a, b | pa − baa − xbabb, pb − babb, class = 4 � ( 0 ≤ x < p ) The groups of order p 7 – p. 12
An example � a, b | pa − baa − xbabb, pb − babb, class = 4 � ( 0 ≤ x < p ) My M AGMA program computes this as a Lie algebra over Z [ x, y, z, x 1 , x 2 , . . . , x 12 ] . The groups of order p 7 – p. 12
An example � a, b | pa − baa − xbabb, pb − babb, class = 4 � ( 0 ≤ x < p ) My M AGMA program computes this as a Lie algebra over Z [ x, y, z, x 1 , x 2 , . . . , x 12 ] . The power map u �→ pu is handled as a linear map from L to L satisfying the relations ( pu ) v = p ( uv ) for all u, v ∈ L . The groups of order p 7 – p. 12
An example � a, b | pa − baa − xbabb, pb − babb, class = 4 � ( 0 ≤ x < p ) My M AGMA program computes this as a Lie algebra over Z [ x, y, z, x 1 , x 2 , . . . , x 12 ] . The power map u �→ pu is handled as a linear map from L to L satisfying the relations ( pu ) v = p ( uv ) for all u, v ∈ L . a 1 = a, a 2 = b a 3 = ba a 4 = baa, a 5 = bab a 6 = babb The groups of order p 7 – p. 12
Computing the automorphism group Consider an automorphism given by a 1 �→ x 1 a 1 + x 2 a 2 + x 3 a 3 + x 4 a 4 + x 5 a 5 + x 6 a 6 a 2 �→ x 7 a 1 + x 8 a 2 + x 9 a 3 + x 10 a 4 + x 11 a 5 + x 12 a 6 The groups of order p 7 – p. 13
Computing the automorphism group Consider an automorphism given by a 1 �→ x 1 a 1 + x 2 a 2 + x 3 a 3 + x 4 a 4 + x 5 a 5 + x 6 a 6 a 2 �→ x 7 a 1 + x 8 a 2 + x 9 a 3 + x 10 a 4 + x 11 a 5 + x 12 a 6 The program gives the following conditions on x 1 , x 2 , . . . , x 12 class by class. The groups of order p 7 – p. 13
Computing the automorphism group Consider an automorphism given by a 1 �→ x 1 a 1 + x 2 a 2 + x 3 a 3 + x 4 a 4 + x 5 a 5 + x 6 a 6 a 2 �→ x 7 a 1 + x 8 a 2 + x 9 a 3 + x 10 a 4 + x 11 a 5 + x 12 a 6 At class 2, nothing. The groups of order p 7 – p. 13
Computing the automorphism group Consider an automorphism given by a 1 �→ x 1 a 1 + x 2 a 2 + x 3 a 3 + x 4 a 4 + x 5 a 5 + x 6 a 6 a 2 �→ x 7 a 1 + x 8 a 2 + x 9 a 3 + x 10 a 4 + x 11 a 5 + x 12 a 6 At class 3: − x 2 1 x 8 + x 1 x 2 x 7 + x 1 = 0 − x 1 x 2 x 8 + x 2 2 x 7 = 0 x 7 = 0 This gives x 2 = x 7 = 0 , x 8 = x − 1 1 . The groups of order p 7 – p. 13
Computing the automorphism group Consider an automorphism given by a 1 �→ x 1 a 1 + x 2 a 2 + x 3 a 3 + x 4 a 4 + x 5 a 5 + x 6 a 6 a 2 �→ x 7 a 1 + x 8 a 2 + x 9 a 3 + x 10 a 4 + x 11 a 5 + x 12 a 6 Set x 2 = x 7 = 0 , and then at class 4 we have − x 2 1 x 8 + x 1 = 0 − xx 1 x 3 8 + xx 1 = 0 − x 1 x 3 8 + x 8 = 0 These relations give x 1 = x 8 = 1 . The groups of order p 7 – p. 13
L , has order p 9 with The p -covering ring, � a 7 = babba a 8 = pa − baa − xbabb a 9 = pb − babb � L 5 is generated by a 7 = babba , and so the immediate descendants of L are � a, b | pa − baa − xbabb − ybabba, pb − babb − zbabba � with class 5 and 0 ≤ y, z < p . The groups of order p 7 – p. 14
If we apply the automorphism a 1 �→ a 1 + x 3 a 3 + x 4 a 4 + x 5 a 5 + x 6 a 6 a 2 �→ a 2 + x 9 a 3 + x 10 a 4 + x 11 a 5 + x 12 a 6 to � L , then babba �→ babba pa − baa − xbabb + ( x 2 pa − baa − xbabb �→ 3 + 2 x 5 ) babba pb − babb �→ pb − babb So we can take y = 0 , and we have p non-isomorphic descendants for each value of x . � a, b | pa − baa − xbabb, pb − babb − zbabba, class = 5 � The groups of order p 7 – p. 15
Apply the Baker-Campbell-Hausdorff formula, and obtain the group relations [ b, a, a ] · [ b, a, b, b ] x · [ b, a, b, b, a ] ( x +1 / 3) a p = b p [ b, a, b, b ] · [ b, a, b, b, a ] z = The groups of order p 7 – p. 16
M AGMA functions for checking results The groups of order p 7 – p. 17
M AGMA functions for checking results Descendants(G:StepSizes:=[s]) — compute immediate descendant of G of order | G | · p s The groups of order p 7 – p. 17
M AGMA functions for checking results Descendants(G:StepSizes:=[s]) — compute immediate descendant of G of order | G | · p s ClassTwo(p,d,s) — count number of d -generator p -class 2 groups of order p d + s The groups of order p 7 – p. 17
M AGMA functions for checking results Descendants(G:StepSizes:=[s]) — compute immediate descendant of G of order | G | · p s ClassTwo(p,d,s) — count number of d -generator p -class 2 groups of order p d + s IsIsomorphic(P ,Q) The groups of order p 7 – p. 17
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