191 research outputs found
Integral Cayley graphs and groups
We solve two open problems regarding the classification of certain classes of
Cayley graphs with integer eigenvalues. We first classify all finite groups
that have a "non-trivial" Cayley graph with integer eigenvalues, thus solving a
problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups
was introduced by Klotz and Sander. These are groups for which every Cayley
graph has only integer eigenvalues. In the second part of the paper, all Cayley
integral groups are determined.Comment: Submitted June 18 to SIAM J. Discrete Mat
On isomorphisms of abelian Cayley objects of certain orders
AbstractLet m be a positive integer such that gcd(m,ϕ(m))=1 (ϕ is Euler's phi function) with m=p1⋯pr the prime power decomposition of m. Let n=p1a1⋯prar. We provide a sufficient condition to reduce the Cayley isomorphism problem for Cayley objects of an abelian group of order n to the prime power case. In the case of Cayley k-ary relational structures (which include digraphs) of abelian groups, this sufficient condition reduces the Cayley isomorphism problem of k-ary relational structures of abelian groups to the prime power case for Cayley k-ary relational structures of abelian groups. As corollaries, we solve the Cayley isomorphism problem for Cayley graphs of Zn (for the specific values of n as above) and prove several abelian groups (for specific choices of the ai) of order n are CI-groups with respect to digraphs
Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers
We consider the problem of testing isomorphism of groups of order n
given by Cayley tables. The trivial n^{log n} bound on the time
complexity for the general case has not been improved over the past
four decades. Recently, Babai et al. (following Babai et al. in SODA
2011) presented a polynomial-time algorithm for groups without abelian
normal subgroups, which suggests solvable groups as the hard case for
group isomorphism problem. Extending recent work by Le Gall (STACS
2009) and Qiao et al. (STACS 2011), in this paper we design a
polynomial-time algorithm to test isomorphism for the largest class of
solvable groups yet, namely groups with abelian Sylow towers, defined
as follows. A group G is said to possess a Sylow tower, if there
exists a normal series where each quotient is isomorphic to Sylow
subgroup of G. A group has an abelian Sylow tower if it has a Sylow
tower and all its Sylow subgroups are abelian. In fact, we are able
to compute the coset of isomorphisms of groups formed as coprime
extensions of an abelian group, by a group whose automorphism group is
known.
The mathematical tools required include representation theory,
Wedderburn\u27s theorem on semisimple algebras, and M.E. Harris\u27s 1980
work on p\u27-automorphisms of abelian p-groups. We use tools from the
theory of permutation group algorithms, and develop an algorithm for a
parameterized versin of the graph-isomorphism-hard setwise stabilizer
problem, which may be of independent interest
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