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Dear GAP Forum members,

Ed Pegg asked:

I've been trying to figure out how to obtain generators for all of

the order 16 groups. I did manage to obtain the Order Distribution for

all of the 16-groups in GAP's library, but it raised some questions.Both SolvableGroup(16,8) and SolvableGroup(16,9) have the same order

distribution -- [1,1] [2,7] [4,8]. I have a book which states that 27

is the first order where this occurs (Budden - The Fascination of

Groups), but I believe GAP more.Anyways, any methods for getting generators or for building the

multiplication tables for these groups would be appreciated. This is

a lead in for building generators for all the order 24, 48, 60 and

120 order groups.

You are well advised to believe GAP (at least this time!!!)

There are in fact even three groups of order 16 with the order

distribution that you quote, namely in addition to the two that you

mention also (16,2) which is the direct product of a cyclic group of

oder 4 with two cyclic groups of oder 2. However if you compute the

subgroup lattices of these three groups you will see that they have

pairwise different subgroup lattices:

(16,2) has as maximal subgroups: one elementary abelian one of order 8

and six abelian ones which are the direct product C4xC2 of a cyclic

group of order 4 by a cyclic group of order 2,

(16,8) has as maximal subgroups three abelian ones C4xC2, three

dihedral groups of order 8 and one quaternion group of order 8,

(16,9) has as maximal subgroups one elementary abelian of order 8 and

two abelian ones C4xC2.

These maximal subgroups you can distinguish by their order

distribution.

However there is even a pair of groups of order 16 which not only have

the same order distribution but even the same subgroup lattice, namely

the direct product of a cyclic group of order 8 by a cyclic group of

order 2 on one hand and the group (16,11) on the other hand. You can

distinguish these only if in addition you consider conjugacy of

subgroups: while in the abelian group of course all subgroups are

normal, in the other one two of its three subgroups of order 2 are

conjugate.

You can best study such properties and get a real impression of the

subgroup structure using XGAP, which allows you to get a picture of

the subgroup lattice drawn on your monitor.

As to your further question about finding generators for small groups:

There are several lists of small groups available in GAP, (the most

recent one going up to order 1000 omitting only orders 512 and 768).

In most cases these give so-called ag-presentations (or polycyclic

presentations) but you can easily convert to permutation groups and

for soluble groups you can get a minimal generating set, as shown in

the following example:

You can fetch the groups from the small groups library. They are

returned as ag groups.

gap> G := SmallGroup( 16, 8 ); 16_8

A minimal generating set can be determined as follows.

gap> G.generators;

[ a, b, c, d ]

gap> MinimalGeneratingSet( G );

[ b, a ]

To convert <G> into a permutation group, you can use the regular

operation of <G> on its elements via right multiplication.

gap> P := Operation( G, Elements( G ), OnRight ); Group( ( 1, 9, 2,10)( 3,12, 4,11)( 5,15, 6,16)( 7,13, 8,14), ( 1, 5)( 2, 6) ( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15), ( 1, 3, 2, 4)( 5, 7, 6, 8) ( 9,11,10,12)(13,15,14,16), ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14) (15,16) )

Incidentally, the same thing is done if you simply call

`PermGroup(G)'.

gap> Q := PermGroup(G); Group( ( 1, 9, 2,10)( 3,12, 4,11)( 5,15, 6,16)( 7,13, 8,14), ( 1, 5)( 2, 6) ( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15), ( 1, 3, 2, 4)( 5, 7, 6, 8) ( 9,11,10,12)(13,15,14,16), ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14) (15,16) ) gap> Q.bijection; InverseMapping( OperationHomomorphism( 16_8, Group( ( 1, 9, 2,10)( 3,12, 4,11) ( 5,15, 6,16)( 7,13, 8,14), ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16) (12,15), ( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16), ( 1, 2)( 3, 4) ( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16) ) ) )

This bijection <Q> -> <G> allows you to map elements back and forth

between <G> and <Q>.

So you see that generating sets for the groups that you intended to

construct are all there already.

I do not think, that building multiplication tables for such groups is

a worthwhile task, you really cannot see much from them. To look for

multiplication tables is perhaps a wrong advice suggested by books

like the one by Budden, which is wonderful in giving many nice

examples of small groups coming in many disguises, but again and again

refers to such multiplication tables. Looking at the subgroup lattice

or even only parts of that usually tells you much more what the group

is like.

Hopes this helps

Joachim Neubueser

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