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Divisibility and structure constraints on automorphism groups of almost perfect 1‐factorizations of
Author(s) -
Ihrig E.,
Petrie E.
Publication year - 1998
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/(sici)1520-6610(1998)6:5<355::aid-jcd5>3.0.co;2-e
Subject(s) - mathematics , combinatorics , automorphism , factorization , cardinality (data modeling) , vertex (graph theory) , permutation group , group (periodic table) , permutation (music) , discrete mathematics , inner automorphism , automorphism group , graph , chemistry , physics , organic chemistry , algorithm , computer science , acoustics , data mining
Let G be a permutation group acting on a set with N elements such that every permutation with more than m fixed points is the identity. It is easy to verify that | G | divides N ( N − 1) ··· ( N − m ). We show that if gcd(| G |, m !) = 1, then | G | divides ( N − i )( N − j ) for some i and j satisfying 0 ≤ i < j ≤ m . We use this to show that any almost perfect 1‐factorization of K 2 n has an automorphism group whose cardinality divides (2 n − i )(2 n − j ) for some i and j with 0 ≤ i < j ≤ 2 as long as n is odd. An almost perfect 1‐factorization (or APOF) is a 1‐factorization in which the union of any three distinct 1‐factors is connected. This result contrasts with an example of an APOF on K 12 given by Cameron which has PSL(2, ℤ 11 ) as its automorphism group [with cardinality 12(11)(5)]. When n is even and the automorphism group is solvable, we show that either G acts vertex transitively and n is a power of two, or | G | divides 2 n − 2 a for some integer a with 2 a dividing 2 n , or else | G | divides (2 n − i )(2 n − j ) for some i and j with 0 ≤ i < j ≤ 2. We also give a number of structure results concerning these automorphism groups. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 355–380, 1998