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The p-quotient Algorithm Marvin Krings November 23, 2016 Marvin Krings The p-quotient Algorithm November 23, 2016 1 / 49

Algorithms p-quotient Algorithm Given a finitely generated group Compute certain quotients: Finite p-groups Each central extension of the previous one by an elementary abelian p-group Each quotient of order p k is quotient of one of them. p-group Generation Algorithm Given a finite p-group Compute central extensions of this by an elementary abelian p-group So-called descendants Application: isomorphism test for finite p-groups Marvin Krings The p-quotient Algorithm November 23, 2016 2 / 49

Contents 1 Power-Commutator Presentations 2 The Lower Exponent-p Central Series 3 The p-covering Group 4 The p-quotient Algorithm 5 Immediate Descendants 6 The p-group Generation Algorithm 7 Testing p-groups for Isomorphism Marvin Krings The p-quotient Algorithm November 23, 2016 3 / 49

Power-Commutator Presentations G = X S presentation of a group Given x, y (X X 1 ), is x = G y? Difficult in general presentations Solution: normal forms and special relators Marvin Krings The p-quotient Algorithm November 23, 2016 5 / 49

Power-Commutator Presentations Definition G group, generating set X = {a 1,..., a n }. Power-commutator presentation of G: presentation G = X S with: 1 power relations: a p i = n k=i+1 aα i,k k for all 1 i n 2 commutator relations: [a j, a i ] = n k=j+1 aα i,j,k k for all 1 i < j n, p prime, α i,k, α i,j,k {0,..., p 1}. Definition Normal forms: words w = a e 1 1 ae 2 2... aen n with 0 e i p 1. Marvin Krings The p-quotient Algorithm November 23, 2016 6 / 49

Collection How to find normal forms? Collection: Replace minimal non-normal subwords n a p i a 1 i k=i+1 aα i,k k a p 1 i i+1 k=n a α i,k k a j a i a i a j n k=j+1 aα i,j,k k Indices larger on right hand side terminates Marvin Krings The p-quotient Algorithm November 23, 2016 7 / 49

Consistency Problem: normal form not unique (inconsistencies) G = X S consistent G = p X Reduction: remove generators to gain consistency ua k, v normal forms for the same element u 1 v = G a k Replace a k by u 1 v Remove a k from X Iterate until consistent Marvin Krings The p-quotient Algorithm November 23, 2016 8 / 49

The Lower Exponent-p Central Series Definition G group, p prime. Lower exponent-p central series: G =: P 0 (G) P 1 (G) P 2 (G)... P i+1 (G) := [P i (G), G]P i (G) p usually P i instead of P i (G) Class of G: smallest c with P c (G) = {1} Marvin Krings The p-quotient Algorithm November 23, 2016 10 / 49

The Lower Exponent-p Central Series Properties of the P i : central: P i G, P i /P i+1 Z(G/P i+1 ) P i (θ(g)) = θ(p i (G)) for homomorphism θ P i char G P i (G/N) = P i (G)N/N for N G G/N of class c P c N. Marvin Krings The p-quotient Algorithm November 23, 2016 11 / 49

p-quotients Properties of the G/P i (assuming G finitely presented): G/P i = (G/Pi+1 )/(P i /P i+1 ) P i /P i+1 : central in G/P i+1, finite elementary abelian p-group G/P i : finite p-group of class i Marvin Krings The p-quotient Algorithm November 23, 2016 12 / 49

The p-covering Group How to compute G/P i+1 from G/P i? Quotient of the p-covering group (G/P i ) Marvin Krings The p-quotient Algorithm November 23, 2016 14 / 49

The p-covering Group Definition p prime, G p-group on d generators, F free group on d generators with G = F /R. p-covering group of G: G := F /([R, F ]R p ). Theorem p, G, G as above, H group on d generators with G = H/Z for some elementary abelian p-group Z which is a central subgroup of H. Then H is a homomorphic image of the p-covering group G, i.e., it is isomorphic to a quotient group of G. Class of G : c + 1 or c Marvin Krings The p-quotient Algorithm November 23, 2016 15 / 49

Enforcing Relations Which quotient is G/P i+1? The largest one fulfilling the relations in R Theorem p prime, G finitely generated group, G = P 0 P 1... lower exponent-p central series of G. Then G/P i+1 is the largest central extension of G/P i by an elementary abelian p-group such that there is an epimorphism G G/P i+1. Marvin Krings The p-quotient Algorithm November 23, 2016 16 / 49

The p-quotient Algorithm Computes consistent power-commutator presentation of G/P i and epimorphism G G/P i Works iteratively: Finds G/P i+1 as a quotient of (G/P i ) and extends the epimorphism G G/P i. Marvin Krings The p-quotient Algorithm November 23, 2016 18 / 49

The p-quotient Algorithm Algorithm (First p-quotient) Let p be a prime and G = X S a group with X = {x 1,..., x k }. 1 Initialize M to be the 0 k matrix over the field F p. 2 For each r S: 1 For each 1 j k, define f j to be the sum of the exponents of x j in r. 2 Add the row (f 1... f k ) to M. 3 Use the Gaussian algorithm and renumbering of the elements of X to compute a basis {b 1,..., b d } of the nullspace of M with the property that (b i ) j = δ ij for all 1 i, j d. 4 Define a set Y := {a 1,..., a d } and a map θ 1 : X (Y ), θ 1 (x j ) := a (b 1) j 1... a (b d ) j d. 5 Define T := {a p i 1 i d} {[a j, a i ] 1 i < j d}. 6 Return Y T and θ i. Marvin Krings The p-quotient Algorithm November 23, 2016 19 / 49

The p-quotient Algorithm Example: Q 8 = i, j, k, ē ē 2, ēi 2, ēj 2, ēk 2, ēijk 0 0 0 2 0 0 0 0 2 0 0 1 M = 0 2 0 1 0 0 2 1 = 0 0 0 1 0 0 0 1 0 0 0 1 F5 4 2 1 1 1 1 1 1 1 1 1 0 ker M = 0 1, 1 1 0 0 Q 8 /P 1 = a1, a 2 [a 2, a 1 ] = ɛ, a 2 1 = ɛ, a 2 2 = ɛ θ 1 : i a 1, j a 2, k a 1 a 2, ē ɛ Marvin Krings The p-quotient Algorithm November 23, 2016 20 / 49

The p-quotient Algorithm a 1,..., a d generate the G/P i. a d+1,..., a n defined by the relations. Definition G = X S power-commutator presentation, X = {a 1,..., a n }, 1 k n. If there is a relation with right hand side wa k for some w {a 1,..., a k 1 }, we call an arbitrary, but fixed one of them the definition of a k. Marvin Krings The p-quotient Algorithm November 23, 2016 21 / 49

The p-quotient Algorithm Algorithm (Tails) Let p be a prime, G a group on d generators and G/P i the i-th p-quotient of G. Furthermore, let X S with X = {a 1,..., a n } be a consistent power-commutator presentation of G/P i. 1 Start with a set X := X of generators and an empty set S of relations. 2 Add all the definitions from S to S. 3 For every other relation (u = v) S: 1 Add a new generator (tail) a k to X. 2 Add a new relation u = va k to S. 3 Add the relation a p k = ɛ to S. 4 Add the relations [a k, a i ] = ɛ, 1 i k 1 to S. 4 Return X S. Marvin Krings The p-quotient Algorithm November 23, 2016 22 / 49

The p-quotient Algorithm Algorithm (Tails) 1... 2 Add all the definitions from S to S. 3 For every other relation (u = v) S: 1 Add a new generator (tail) a k to X. 2 Add a new relation u = va k to S. 3 Add the relation a p k = ɛ to S. 4 Add the relations [a k, a i ] = ɛ, 1 i k 1 to S. 4... Factoring out new generators old presentation New generators are central and of order p. New generators = old non-definition relations = R/R X S = G. Marvin Krings The p-quotient Algorithm November 23, 2016 23 / 49

The p-quotient Algorithm Example: Q 8 /P 1 = a1, a 2 [a 2, a 1 ] = ɛ, a1 2 = ɛ, a2 2 = ɛ (Q 8 /P 1 ) = a1,..., a 5 [a 2, a 1 ] = a 3, a1 2 = a 4, a2 2 = a 5 Presentation is consistent. Marvin Krings The p-quotient Algorithm November 23, 2016 24 / 49

The p-quotient Algorithm Algorithm (Relation Enforcement) Let p be a prime, G a finitely generated group and G = P 0 P 1... the lower exponent-p central series of G. Furthermore, let X S with X = {x 1,..., x k } be a presentation of G, Y T with Y = {a 1,..., a n } be a consistent power-commutator presentation of G/P i for some i, such that {a 1,..., a d } is a minimal generating set for some d, Y T with Y = {a 1,..., a m } be the consistent power-commutator presentation of (G/P i ) that is obtained by applying the tails algorithm and the consistency algorithm to Y T and θ i : X (Y ) be a map with the property θ(x j ) = a j for all 1 j d that induces an epimorphism G G/P i. Marvin Krings The p-quotient Algorithm November 23, 2016 25 / 49

Algorithm (Relation Enforcement) 1 Initialize M to be the 0 (k d + m n) matrix over the field F p and initialize θ i+1 : X (Y ) with θ i+1 (x) := θ i (x) for all x X. 2 For each r S: 1 Compute θ i (r) and collect it with respect to Y T, yielding a en+1 m. n+1... aem 2 For each 1 j k, define f j to be the sum of the exponents of x j in r. 3 Add the row (f d+1... f k e n+1... e m ) to M. 3 Replace M by its row-reduced echelon form. 4 For each row (0... 0 ẽ n+1... ẽ m ) of M: Do a reduction of Y T with the words ɛ and aẽn+1 n+1... aẽm m. Change the right hand side of the matrix accordingly. 5 For each remaining row (0... 01... ẽ n+1... ẽ m ) with 1 j k being the number of the first non-zero entry: Redefine θ i+1 (x j ) := θ i (x j )aẽn+1 n+1... aẽm m. 6 Return Y T and θ i+1. Marvin Krings The p-quotient Algorithm November 23, 2016 26 / 49

The p-quotient Algorithm New generators form a vector space aẽn+1 n+1... aẽm m from step 4 in image of no extension of θ i S ker θ i+1 relations fulfilled Reduction maintains consistency. Marvin Krings The p-quotient Algorithm November 23, 2016 27 / 49

The p-quotient Algorithm Example: (Q 8 /P 1 ) = a1,..., a 5 [a 2, a 1 ] = a 3, a 2 1 = a 4, a 2 2 = a 5 M = θ 1 : i a 1, j a 2, k a 1 a 2, ē ɛ θ 1 (ē 2 ) = ɛ 2 = ɛ θ 1 (ēi 2 ) = ɛa 2 1 = a 4 θ 1 (ēj 2 ) = ɛa 2 2 = a 5 θ 1 (ēk 2 ) = ɛa 1 a 2 a 1 a 2 = a 3 a 4 a 5 θ 1 (ēijk) = ɛa 1 a 2 a 1 a 2 = a 3 a 4 a 5 0 2 0 0 0 0 1 0 1 0 0 1 0 0 1 2 1 1 1 1 1 1 1 1 1 = 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 F5 5 2 Marvin Krings The p-quotient Algorithm November 23, 2016 28 / 49

The p-quotient Algorithm Example: factor out a 3 a 5 : factor out a 3 a 4 : (Q 8 /P 1 ) = a1,..., a 5 [a 2, a 1 ] = a 3, a 2 1 = a 4, a 2 2 = a 5 θ 1 : i a 1, j a 2, k a 1 a 2, ē ɛ 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 a 1, a 2, a 3, a 4 [a 2, a 1 ] = a 3, a 2 1 = a 4, a 2 2 = a 3 a 4 a 5 a 3 a 4 Q 8 /P 2 = a1, a 2, a 3 [a 2, a 1 ] = a 3, a 2 1 = a 3, a 2 2 = a 3 Marvin Krings The p-quotient Algorithm November 23, 2016 29 / 49

The p-quotient Algorithm Example: (Q 8 /P 1 ) = a1,..., a 5 [a 2, a 1 ] = a 3, a 2 1 = a 4, a 2 2 = a 5 θ 1 : i a 1, j a 2, k a 1 a 2, ē ɛ Q 8 /P 2 = a1, a 2, a 3 [a 2, a 1 ] = a 3, a 2 1 = a 3, a 2 2 = a 3 ( 1 0 0 0 1 1 ) θ 2 : {i, j, k, ē} {a 1, a 2, a 3 }, i a 1, j a 2, k a 1 a 2, ē a 3 Presentation consistent Q 8 /P 2 = 2 3 = 8 = Q 8 Q 8 /P 2 = Q8 P 2 = {1}, Q 8 has class 2. Marvin Krings The p-quotient Algorithm November 23, 2016 30 / 49

The p-quotient Algorithm Algorithm (p-quotient) Let p be a prime, G = X S with X = {x 1,..., x k } and G = P 0 P 1... the lower exponent-p central series of G. 1 Compute a presentation Y 1 T 1 of the first p-quotient G/P 1 and an epimorphism θ 1 : G G/P 1. 2 For i N: 1 Compute a power-commutator presentation Yi p-covering group (G/P i ) using the tails algorithm. 2 Make Y T i for the i Ti consistent. 3 Enforce the relations in S to compute a consistent power-commutator presentation Y i+1 T i+1 of the p-quotient G/P i+1 epimorphism θ i+1 : G G/P i+1. 4 If Y i+1 = Y i, stop. and an Marvin Krings The p-quotient Algorithm November 23, 2016 31 / 49

Immediate Descendants Given a finite p-group G, look for extensions Central extensions by elementary-abelian p-groups Use G again Marvin Krings The p-quotient Algorithm November 23, 2016 33 / 49

Immediate Descendants Definition p prime, G finite p-group on d generators with class c, F free group on d generators with G = F /R. G := F /R with R := [R, F ]R p : p-covering group of G. 1 p-multiplicator: R/R 2 nucleus: P c (G ) 3 descendant: group H on d generators with G = H/P c (H) 4 immediate descendant: descendant with class c + 1 5 allowable subgroup: M/R R/R with H = (F /R )/(M/R ) for an immediate descendant H Marvin Krings The p-quotient Algorithm November 23, 2016 34 / 49

Immediate Descendants Theorem A proper subgroup M/R R/R of the p-multiplicator is allowable if and only if P c (G )(M/R ) = R/R. Marvin Krings The p-quotient Algorithm November 23, 2016 35 / 49

Immediate Descendants Determine all allowable subgroups all immediate descendants Possibly redundant list: isomorphic immediate descendants Isomorphism can be extended to automorphism of G. Induces permutation of the allowable subgroups Orbit of permutations isomorphism type Marvin Krings The p-quotient Algorithm November 23, 2016 36 / 49

Immediate Descendants Theorem Let M 1 /R and M 2 /R be allowable subgroups and ϕ: F /M 1 F /M 2 be an isomorphism. Then there exists an automorphism α : F /R F /R such that α (M 1 /R ) = M 2 /R, α (R/R ) = R/R, α induces ϕ on F /M 1 = (F /R )/(M 1 /R ), The restriction of α to R/R is uniquely determined by ϕ. Marvin Krings The p-quotient Algorithm November 23, 2016 37 / 49

Immediate Descendants Corollary Let α: F /R F /R be an automorphism. Then there exists an automorphism α : F /R F /R such that α (R/R ) = R/R, α induces α on F /R = (F /R )/(R/R ), The restriction of α to R/R is uniquely determined by α. Marvin Krings The p-quotient Algorithm November 23, 2016 38 / 49

Immediate Descendants Lemma Let α : F /R F /R be an automorphism as obtained by the theorem or the corollary above. Then α induces a permutation α of the allowable subgroups. Theorem {α α Aut(G)} is a group. Let M 1 /R and M 2 /R be two allowable subgroups. F /M 1 is isomorphic to F /M 2 if and only if M 1 /R and M 2 /R lie in the same orbit of the permutation group. Marvin Krings The p-quotient Algorithm November 23, 2016 39 / 49

The p-group Generation Algorithm Input: finite p-group G Compute immediate descendants How to extend the automorphisms of G? Marvin Krings The p-quotient Algorithm November 23, 2016 41 / 49

The p-group Generation Algorithm Algorithm (Extend Automorphism) p prime, G = X S consistent power-commutator presentation, X S result of the tails and consistency algorithms, α: X (X ) map that induces automorphism on G. For each a i X \ X : 1 Define w := uv 1, where u = va i is the definition of a i. 2 Apply the map α to w. 3 Collect the result with respect to X S, yielding a word w. 4 Define α (a i ) := w. Marvin Krings The p-quotient Algorithm November 23, 2016 42 / 49

The p-group Generation Algorithm Algorithm (p-group Generation) p prime, G = X S with X = {a 1,..., a n } consistent power-commutator presentation, Aut(G) = α 1,..., α m. 1 Compute p-covering group G = X S using the tails and consistency algorithms. 2 Compute p-multiplicator X \ X S G and nucleus P c (G ) = X N S G, where c is the class of G. 3 Compute allowable subgroups M 1 = Y1 S,..., M l = Yl S. 4 For each 1 i m: 1 Compute α i. 2 Compute the permutation α i of the allowable subgroups induced by α i. 5 Compute the orbits of α 1,..., α m on the allowable subgroups. 6 For each orbit: 1 Choose a representative M = Y S of the orbit. 2 Construct the immediate descendant H = G /M. Marvin Krings The p-quotient Algorithm November 23, 2016 43 / 49

The p-group Generation Algorithm Example: D 8 = a1, a 2, a 3 [a 2, a 1 ] = a 3, a 2 1 = a 3 (D 8 ) = a1,..., a 6 [a 2, a 1 ] = a 3, [a 3, a 2 ] = a 4, a 2 1 = a 3 a 5, a 2 2 = a 6, a 2 3 = a 4 R/R = a 4, a 5, a 6, P 2 (D8) = a 4 Allowable subgroups have dimension 2. Possible: a 4, a 5, a 4, a 6, a 5, a 6, a 4, a 5 a 6, a 5, a 4 a 6, a 6, a 4 a 5, a 4 a 5, a 5 a 6 Actually allowable: a 5, a 6, a 5, a 4 a 6, a 6, a 4 a 5, a 4 a 5, a 5 a 6 Marvin Krings The p-quotient Algorithm November 23, 2016 44 / 49

The p-group Generation Algorithm (D 8 ) = a1,..., a 6 [a 2, a 1 ] = a 3, [a 3, a 2 ] = a 4, a 2 1 = a 3 a 5, a 2 2 = a 6, a 2 3 = a 4 Aut(D 8 ) = α 1, α 2, α 1 : a 1 a 1, a 2 a 1 a 2, a 3 a 3 α 2 : a 1 a 1 a 3, a 2 a 2, a 3 a 3 α 1(a 4 ) := α 1 (a 3 ) 2 = a 2 3 = a 4 α 1(a 5 ) := α 1 (a 1 ) 2 α 1 (a 3 ) 1 = a 2 1a 1 3 = a 5 α 1(a 6 ) := α 1 (a 2 ) 2 = (a 1 a 2 ) 2 = a 5 a 6 α 2(a 4 ) := α 2 (a 3 ) 2 = a 2 3 = a 4 α 2(a 5 ) := α 2 (a 1 ) 2 α 2 (a 3 ) 1 = (a 1 a 3 ) 2 a 1 3 = a 4 a 5 α 2(a 6 ) := α 2 (a 2 ) 2 = a 2 2 = a 6 Marvin Krings The p-quotient Algorithm November 23, 2016 45 / 49

The p-group Generation Algorithm (D 8 ) = a1,..., a 6 [a 2, a 1 ] = a 3, [a 3, a 2 ] = a 4, a 2 1 = a 3 a 5, a 2 2 = a 6, a 2 3 = a 4 allowable: a 5, a 6, a 5, a 4 a 6, a 6, a 4 a 5, a 4 a 5, a 5 a 6 α 1 : a 4 a 4, a 5 a 5, a 6 a 5 a 6 α 2 : a 4 a 4, a 5 a 4 a 5, a 6 a 6 α2( a 5, a 6 ) = a 6, a 4 a 5 α1( a 6, a 4 a 5 ) = a 4 a 5, a 5 a 6 α2( a 4 a 5, a 5 a 6 ) = a 5, a 4 a 6 all allowable subgroups in the same orbit. Factor out a 5, a 6 : a 1, a 2, a 3, a 4 [a 2, a 1 ] = a 3, [a 3, a 2 ] = a 4, a1 2 = a 3, a3 2 = a 4 is the only immediate descendant (up to isomorphism). Marvin Krings The p-quotient Algorithm November 23, 2016 46 / 49

Testing p-groups for Isomorphism Combination of p-quotient and p-group generation Compute standard presentation of a finite p-group Isomorphic Groups equal standard presentations Marvin Krings The p-quotient Algorithm November 23, 2016 48 / 49

Algorithm (Standard Presentation) p prime, G finite p-group of class c. 1 Compute a consistent power-commutator presentation Y 1 T 1 of G/P 1 and an epimorphism θ i : G Y 1 T 1. 2 For each 0 i < c: 1 Compute a consistent power-commutator presentation Y i T i of (G/P i ) and a consistent power-commutator presentation for each allowable subgroup of G/P i. 2 Compute a consistent power-commutator presentation Y i+1 T i+1 of G/P i+1 as well as an epimorphism θ i+1 : G Y i+1 T i+1. Let w 1,..., w n Ti with w 1,..., w n words in Yi \ Y i be the subgroup factored out in step 4 of the relation enforcement algorithm. 3 Check which of the allowable subgroups is equal to w 1,..., w n Ti. 4 Define Y i+1 T i+1 as the consistent power-commutator presentation returned by the p-group generation algorithm for the corresponding immediate descendant. Let α be the automorphism of Yi maps Y i+1 T i+1 to Y i+1 T i+1. 5 Define θ i+1 := α θ i+1. T i that Marvin Krings The p-quotient Algorithm November 23, 2016 49 / 49

George Havas and M. F. Newman, Application of computers to questions like those of Burnside, Burnside Groups, (Bielefeld, 1977). Lecture Notes in Math., 806, 211-230. Springer-Verlag, Berlin, Heidelberg, New York, 1980. Derek F. Holt, Bettina Eick and Eamonn A. O Brien, Handbook of computational group theory. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D.C., 2005. M. F. Newman, Werner Nickel and Alice C. Niemeyer, Descriptions of groups of prime-power order, J. Symbolic Comput., 25, 665-682, 1998. M. F. Newman and E. A. O Brien, Application of computers to questions like those of Burnside, II, Internat. J. Algebra Comput., 6, 593-605, 1996. E. A. O Brien, Isomorphism testing for p-groups, J. Symbolic Comput., 17, 133-147, 1994. Michael Vaughan-Lee, An Aspect of the Nilpotent Quotient Algorithm, Computational Group Theory, (Durham, 1982), 76-83. Academic Press, London, New York, 1984. Marvin Krings The p-quotient Algorithm November 23, 2016 49 / 49