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##

Automorphisms

An *automorphism* of a block design is a permutation of the set
of points of the design such that, if this permutation is applied to the
elements of each block, the multiset of blocks is the same as before.
(In other words: the block multiset is a list of lists; if we apply the
permutation to all elements of the inner lists, re-sort each inner list,
and then re-sort the outer list, the result is the same as the original
list.)

The collection of all automorphisms forms a *group*, that is, it
is closed under composition of permutations. Thus, the automorphism
group of a design is a permutation group on the set of points.

If the block design does not have repeated blocks, then each
automorphism induces a permutation on the set
of
block indices: this permutation carries to if the image of the
-th block under the automorphism is the -th block. In this case,
the automorphism group has an induced action on the set of block
indices. If there are repeated blocks, the action on the set of block
indices is undefined.

For example, the example in the Introduction has an automorphism
(mapping 0 to 1, 1 to 3, etc.) Altogether this famous
design has 168 automorphisms.

The specifications for automorphism groups and their properties for
block designs are:

block_design_automorphism_group = element automorphism_group {
permutation_group,
block_design_automorphism_group_properties ?
}

block_design_automorphism_group_properties = element automorphism_group_properties {
element block_primitive {
attribute flag { "true" | "false" | "not_applicable" }
} ?
,
element no_block_orbits {
attribute value { xsd:positiveInteger | "not_applicable" }
} ?
,
element degree_block_transitivity {
attribute value { xsd:nonNegativeInteger | "not_applicable" }
} ?
}

Permutation groups and their properties have already been described in
section 5.
Some properties of the automorphism group are specific to block
designs, and are (optionally) described separately under
`automorphism_group_properties`. They are:

`block_primitive`

True if the group acts primitively on blocks. (If there are repeated
blocks, this is not defined, and takes the value
`not_applicable`.)

`no_block_orbits`

The number of orbits on blocks. (If there are repeated blocks, this is
not defined, and takes the value `not_applicable`.)

`degree_block_transitivity`

The maximum number such that the group is -transitive
on blocks. (If there are repeated blocks, this is not defined, and takes
the value `not_applicable`.)

** Next:** Resolutions
** Up:** Block Designs
** Previous:** -wise balance
** Contents**
Peter Dobcsanyi
2003-12-15