Year: 1974
Type: article
Abstract: Let G be a transitive permutation group on a finite set \Omega .Let G_{a} be the stabilizer of a\in\Omega in G. Let \Delta_{1}=\{a\} , \Delta_{2} , \cdots , \Delta_{r} be the orbits of G_{a} on \Omega (these are called the suborbits of (G, \Omega )).Then we say that the pemuta- tion group (G, \Omega) is of rank r, and we call |\Delta_{i}| 's the subdegrees of (G, \Omega) (From the transitivity of (G, \Omega) , the |\Delta_{i}| 's are independent of the choice of a\in\Omega) .When (G, \Omega) is given, it is sometimes required to obtain the sub- degrees.The purpose of this short note is to give a practical method to calculate the subdegrees when the structure of the group G is fairly known.The authors thank the members of "YUGENGUN TANSHIN" (a semi- private set of letters written in Japanese circulated periodically among the Japanese young people who are studying finite group theory), especially Mr. Hikoe Enomoto and Miss Yoko Usami, for valuable discussions.In par- ticular, Usami calculated some important examples by applying our method (see Appendix).NOTATION: Let G be a transitive permutation group on a set \Omega , and let H=G_{a} be the stabilizer of a\in\Omega .Let \mathfrak{H}_{1} , \mathfrak{H}_{2} , \cdots , \mathfrak{H}_{t} be the sets of all H -conjugate subgroups of H(i.e., any subgroup X\leqq H is contained in some and only one \mathfrak{H}_{i} ).Moreover we fix an element H_{i}\in \mathfrak{H}_{i} (i=1,2, \cdots, t) .Let us define a partial order among \mathfrak{H}_{\vee}i 's by \mathfrak{H}_{i}\leqq \mathfrak{H}_{f} if there exist subgroups X_{i}\in \mathfrak{H}_{i} and X_{f}\in \mathfrak{H}_{J} such that X_{i}\leqq X_{f} .If X_{i\neq}<X_{f} , we denote by \mathfrak{H}_{i}<\mathfrak{H}_{J} .Let us set \Omega_{i}=H_{i}\backslash H (the right cosets of H by H_{i} ), then H acts on \Omega_{i} naturally.Let us set I_{\rho}(H_{i}) -{ b\in\Omega|b^{h}=b for any h\in H_{i} } , and I_{9_{f}}(H_{i})= { b\in\Omega_{f}|b^{h}=b for any h\in H_{i} } (Note that the cardinality of these sets are independent of the choice of H_{i} in \mathfrak{H}_{i} and of the choice of H_{f} in \mathfrak{H}_{f},\cdot ) Moreover let us set A_{G,H}(H_{i})= { X\leqq H| there exists g\in G such that X^{g}=H_{i} }(where X^{g}=g^{-1}Xg ), and A_{H,H}(H_{i})=\mathfrak{H}_{i}r *) This is a reproduction of some articles in [2] (1972).
Source: Hokkaido Mathematical Journal
Authors: Eiichi BANNAI, Shiro IWASAKI
Institutions: The University of Tokyo, Hokkaido University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funder: Hokkaido University
OpenAlex ID: W2062933373