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3 The organsation of the data

Sections

  1. The p-group generation algorithm
  2. The groups of order 6561
  3. The groups with order the seventh power of a prime

We include some brief comments on the organisation of the data. As a preliminary step we briefly recall the p-group generation algorithm.

3.1 The p-group generation algorithm

The p-group generation algorithm was developed and implemented by Eamonn O'Brien, and we refer the reader to OBrien90 for a detailed description of the algorithm.

For a brief overview, let P be a p-group. The algorithm uses the lower p-central series, defined recursively by lambda1(P)=P and lambdai+1(P)=[lambdai(P),P]lambdai(P)p for i≥1. The p-class of P is the length of this series. Each p-group P, apart from the elementary abelian ones, is an immediate descendant of the quotient P/R where R is the last non-trivial term of the lower p-central series of P.

Thus all the groups with order 38, except the elementary abelian one, are immediate descendants of groups with order 3k for some k smaller than 8. All of the immediate descendants of a p-group Q are quotients of a certain extension of Q; the isomorphism problem for these descendants is equivalent to the problem of determining orbits of certain subgroups of this extension under an action of the automorphism group of Q. Not all p-groups have immediate descendants, those that do are called capable, and those which do not are called terminal.

O'Brien and Vaughan-Lee's classification of the groups of order p7 in OVL05 is based on a classification of the nilpotent Lie rings of order p7, and the groups of order p7 are obtained from the Lie rings using the Baker-Campbell-Hausdorff formula. O'Brien and Vaughan-Lee classified the nilpotent Lie rings of order p7 using the nilpotent Lie ring generation algorithm, which is a direct analogue of the p-group generation algorithm.

Thus the databases of nilpotent Lie rings of order p7 and of the groups of order 38 are organized according to these algorithms: the immediate descendants of order p7 of each nilpotent Lie ring of order less that p7 are grouped together in the database of nilpotent Lie rings, and the immediate descendants of order 38 of each group of order less than 38 are grouped together in the database of groups of order 38.

3.2 The groups of order 6561

The database of groups of order 38 is organized according to rank and p-class. Here rank is the rank of the Frattini quotient, i.e. the size of a minimal generating set, and p-class is as defined in the previous section. The following table gives the number of groups of order 38 of each rank and p-class, with the (i,j) entry corresponding to rank i and p-class j.

  | 1   2        3        4       5      6     7   8
--|-------------------------------------------------
1 | 0   0        0        0       0      0     0   1 
2 | 0   0        58       486     1343   330   9   0 
3 | 0   4        216747   40521   2163   24    0   0 
4 | 0   23361    494666   22343   51     0     0   0
5 | 0   578478   14796    80      0      0     0   0 
6 | 0   566      39       0       0      0     0   0 
7 | 0   10       0        0       0      0     0   0 
8 | 1   0        0        0       0      0     0   0

In the list of all groups of order 38 the first group is the cyclic group, then the 2-generator groups follow in order of increasing p-class and so on. The above table can thus be used to find the range of numbers of groups with a given rank and p-class.

As mentioned above, the database is organized according to the p-group generation algorithm. For example, the 9 groups of rank 2, p-class 7, and order 38 are numbered from 2219--2227. The groups numbered 2219 and 2220 are descendants of SmallGroup(37,384), and the groups numbered 2221--2227 are descendants of SmallGroup(37,386). Similarly, the 24 groups of rank 3, p-class 6, and order 38 are numbered from 261663--261686. The first four of these groups are descendants of SmallGroup(37,5841), the next 17 are descendants of SmallGroup(37,5844), and the last 4 are descendants of SmallGroup(37,5849).

3.3 The groups with order the seventh power of a prime

The groups of order p7 for p>11 are obtained from the LiePRing database of nilpotent Lie rings of order p7 using Willem de Graaf's implementation of the Baker-Campbell-Hausdorff formula. The LiePRing database is organized according to the output from the nilpotent Lie ring generation algorithm. For any given p the first Lie ring in the database is the cyclic Lie ring of order p7. Next come the two generator Lie rings, then the three generator Lie rings, and so on, ending with the six generator Lie rings, and then finally the elementary abelian Lie ring of rank 7. The first four of the two generator nilpotent Lie rings of order p7 are immediate descendants of the Lie ring

< a,b|pb,class 3 >

of order p4. The next p2+8p+25 are immediate descendants of the Lie ring

< a,b|baa,bab,pba,class 3 >

of order p5, and the next p+6+(p2+3p+10) gcd(p-1,3) are immediate descendants of

< a,b|babb,pa,pb,class 4 >.

And so on. The nine rank 6 Lie rings in the database are the rank 6, p-class 2 Lie rings. These are the immediate descendants of the elementary abelian Lie ring of rank 6.

There is a complete list of presentations for the nilpotent Lie rings of order pk for k≤7 valid for all p>3 in the document p567.pdf supplied with the documentation for the LiePRing package. The presentations are grouped as described above, with each group of presentations giving the immediate descendants of a Lie ring of smaller order.

In a few cases the descendants of a parametrized family of Lie rings are grouped together.

There is no easy way to determine the numbering of (say) the three generator Lie rings of p-class 4 since the numbers depend on p in a very complicated way, and generally there is no easy efficient way of searching the database for a group with particular properties. In view of the numbers of groups of order p7 and the time needed to generate a complete list, this means that the database will be of limited use for most people. A user who wants to access a particular batch of descendants as described in p567.pdf is advised to use the LiePRing package directly, as this package also has an option to obtain the corresponding groups via Willem de Graaf's implementation of the Baker-Campbell-Hausdorff formula. On the other hand, there are only

2p3+13p2+64p+145+(p2+10p+56) gcd(p-1,3)+(4p+28) gcd(p-1,4)

+(2p+12) gcd(p-1,5)+ gcd(p-1,7)+4 gcd(p-1,8)+ gcd(p-1,9)

two generator groups of order p7 for p>5 and a complete list of them can be generated quite quickly for moderate values of p. The table below gives the number of d generator groups of order p7 valid for all p>5, and the user can use the table to compute the range of numbers needed to access the d generator groups of order p7 for any given p.

rank 1
1

rank 2
2p3+13p2+64p+145+(p2+10p+56)gcd (p-1,3) +(4p+28)gcd (p-1,4)
+(2p+12)gcd (p-1,5)+gcd (p-1,7)+4gcd (p-1,8)+gcd (p-1,9)

rank 3
2p5+9p4+29p3+99p2+380p+1100+(3p2+28p+189)gcd (p-1,3)
+(p2+13p+84)gcd (p-1,4)+(p+17)gcd (p-1,5)+gcd (p-1,8)+3gcd (p-1,7)

rank 4
p5+3p4+13p3+57p2+248p+1044+(6p+46)gcd (p-1,3)
+(2p+23)gcd(p-1,4)+2gcd(p-1,5)

rank 5
p2+15p+155

rank 6
9

rank 7
1

The total number of groups of order p7 for all p > 5 is given by

3p5+12p4+44p3+170p2+707p+2455

+(4p2+44p+291)gcd (p-1,3)+(p2+19p+135)gcd (p-1,4)

+(3p+31)gcd (p-1,5)+4gcd (p-1,7)+5gcd (p-1,8)+gcd (p-1,9).

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sglppow manual
Dezember 2014