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 $[0, \ldots , b-1]$ of block indices: this permutation carries $i$ to $j$ if the image of the $i$-th block under the automorphism is the $j$-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 $[1,3,5,2,0,6,4]$ (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 $s$ such that the group is $s$-transitive on blocks. (If there are repeated blocks, this is not defined, and takes the value not_applicable.)