We introduce an invariant of finite permutation groups called the arity which is well known to model theorists but has not been examined from an algebraic point of view. There are few cases in which this invariant is known explicitly. We analyze the behavior of this invariant in power representation
T-sets and permutation groups
β Scribed by Michael Aschbacher
- Publisher
- Elsevier Science
- Year
- 1974
- Tongue
- English
- Weight
- 947 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0021-8693
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π SIMILAR VOLUMES
We show that every group is the full automorphism group of an unordered k-relation for given k 2 2 on some suitable set. We are also concerned with the cardinality of a minimal representation with this property as a function of k.
A G-loop is a loop which is isomorphic to all its loop isotopes. We apply some theorems about permutation groups to get information about G-loops. In particular, we study G-loops of order pq, where p < q are primes and p q -1 . In the case p = 3, the only G-loop of order 3q is the group of order 3q.
If a sequence of transitive permutation groups G of degree n have orders which are not too large (log IGI--o(n~) suttices), then the number of orbits on the power set is asymptotically 2n/]GI, and almost all of these orbits are regular. This conclusion holds in particular for primitive groups.
## AstiC-Vidal, A. and V. Dugat, Near-homogeneous tournaments and permutation groups, Discrete Mathematics 102 (1992) 111-120.