19/10/19 19:25:01.03 ti2BclkQ.net
S5の位数20の部分群
URLﾘﾝｸ(groupprops.subwiki.org)(1,5)
General affine group:GA(1,5)
(抜粋)
As GA(1,q), q = 5: q(q - 1) = 5(5 - 1) = 20
As holomorph of cyclic group:Z5: |Z5||Aut(Z5)| = 5・4 = 20
As Sz(q), q = 2: q^2(q^2 + 1)(q - 1) = 2^2(2^2 + 1)(2 - 1) = 4・5・1 = 20
Group properties
Function Value
abelian group No
nilpotent group No
metacyclic group Yes
supersolvable group Yes
solvable group Yes
Frobenius group Yes
Camina group Yes
URLﾘﾝｸ(people.maths.bris.ac.uk)
Tim Dokchitser Arithmetic/Algebraic Geometry University of Bristol
URLﾘﾝｸ(people.maths.bris.ac.uk)
G = F5? order 20 = 2^2・5 Frobenius group Tim Dokchitser
URLﾘﾝｸ(groupprops.subwiki.org)
General affine group of degree one
GA(1,K) = K semix K^*
URLﾘﾝｸ(ja.wikipedia.org)
アフィン群
URLﾘﾝｸ(en.wikipedia.org)
Frobenius group
(抜粋)
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.
Structure
A subgroup H of a Frobenius group G fixing a point of the set X is called the Frobenius complement.
The identity element together with all elements not in any conjugate of H form a normal subgroup called the Frobenius kernel K.
(This is a theorem due to Frobenius (1901); there is still no proof of this theorem that does not use character theory, although see [1].)
The Frobenius group G is the semidirect product of K and H:
G=K semix H
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