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Dear GAP-Forum,

Igor Schein asked (Jan 18):

I'd like to be able to determine in GAP4 whether a group has a

quarternion subgroup. For a example, I know apriori that

SmallGroup(36,29) does have one. I would like to visualize that in

GAP.

The example mentioned looks odd: A group of order 36 cannot have a

quaternion subgroup of order 8, since 8 does not divide 36, moreover

there are only 14 (isomorphism classes of) groups of order 36.

However the question does make sense of course and here is some advice

on it:

There is no special function in GAP which would directly answer the

question if a given group contains a subgroup isomorphic to the

quaternion group, or that would just find such a subgroup.

If the 'table of marks' of the group in question is stored in the

library of table of marks, then the question can be answered from

inspection of that table. See the respective chapter in the GAP

Reference Manual.

However this will only be the case for a limited number of groups and

it would not be advisable for this purpse to ask GAP to compute the

table of marks for a group for which it is not stored, since doing

this automatically would in fact entail the calculation of the whole

subgroup lattice.

Rather you should write a little GAP function for the purpose, that

could e.g. do the following:

For a given Group G find a 2-Sylow subgroup H.

In H find representatives of the H-conjugacy classes of elements of oder 4.

For each of them, say g, search for elements h in H

- of order 4

- normalizing <g>

- not centralizing g

- wth g^2 = h^2.

<g,h> will then be a quaternion subgroup and if no such pair is found

this way, G will not contain a quatenion subgroup.

Depending on how big the Sylow sbgroup is in comparison with G, other

ways to proceed may even be more advantageous, but in any case all

this can easily be done in GAP.

Finally, to visualize the embedding of such a quaternion group in the

given group, the package XGAP provides the possibility to get a

drawing of the subgroup lattice or parts thereof on the screen.

Hope this helps, if there are further, in particular technical

questions, please write to 'gap-trouble' rather than to 'gap-forum'.

Joachim Neubueser

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