AutGroupStructure(A)
The generating set of the aut group returned by AutGroupSagGroup is
closely related to a particular subnormal series of the aut group. This
function displays a description of the factors of this series.
Let A be the aut
The function
The factor of weight i is A_i/A_{i+1}. A factor of even weight is an
elementary abelian group, and it is described by giving its order. A factor
of odd weight is described by giving a generating set for a faithful
representation of it as a matrix group acting on a layer of the LG-series
of G (the weight 2i-1 factor acts on the LG-series layer
G_i/G_{i+1}).
inputOut.Structure
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As mentioned earlier, each generator of the aut group of G. Let G=G_1 > G_2 > ldots > G_m >
G_{m+1}=1 be the LG-series of G (see More about Special Ag Groups).
For 0 leq i leq m let A_{2i+1} be the subgroup of A containing all
those auts which induce the identity on G/G_{i+1}. Clearly A_1 = A
and A_{2m+1} = 1. Furthermore, let A_{2i+2} be the subgroup of
A_{2i+1} containing those auts which also act trivially on the quotient
G_i / G_{i+1}. Note that A_2/A_3 is always trivial. Thus the subnormal
series
A = A_1 geq A_2 geq ldots geq A_2m+1 = 1
of A is obtained. The subgroup A_i is the weight i subgroup of A.
The weight of a generator alpha of A is defined to be the least i
such that alpha in A_{i}.
AutGroupStructure takes as input an aut group A computed
using AutGroupSagGroup and prints out a description of the non-trivial
factors of the subnormal series of the aut group A.
group has its weight
stored in the record component weight.
inputOut.Weights
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Note that the subgroup A_i of A is generated by the elements of the generating set of A whose weights are at least i. Hence, in analogy to strong generating sets of permutation groups, the generating set of A is a strong generating set relative to the chain of subgroups A_i.
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The generating set of a matrix group displayed by AutGroupStructure
corresponds directly to the list of elements of the corresponding weight in
A.generators. In the example above, the first matrix listed at weight 5
corresponds to A.generators[3], and the last matrix listed at weight 5
corresponds to A.generators[9].
It is also worth noting that the generating set for an aut
group returned
by AutGroupSagGroup can be heavily redundant. In the example given above,
the weight 5 matrix group can be generated by just three of the seven
elements listed (for example elements 1, 5 and 6). The other four elements
can be discarded from the generating set for the matrix group, and the
corresponding elements of the generating set for A can also be discarded.
April 1997