Subnormal divisor

In the group theory , a is sub-group of a group as a subnormal subgroup (or subnormal subgroup hereinafter), if a Subnormalreihe of to exist, that is, if there is a finite chain  ≤ ... ≤  of subgroups of there, so that each normal subgroup of is. ${\displaystyle S}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle S}$ ${\displaystyle S=S_{0}}$ ${\displaystyle S_{k}=G}$ ${\displaystyle G}$ ${\displaystyle S_{i}}$ ${\displaystyle S_{i+1}}$

Subnormal divisors - still under the name of post-invariant subgroup - were first considered by Helmut Wielandt in his post- doctoral thesis, A generalization of invariant subgroups , published in 1939 . Among other things, Wielandt was able to show that in finite groups the product of two subnormal divisors is always subnormal again , that is, the subnormal divisors form an association .

The concept of the subnormal divisor is a generalization of the concept of the normal divisor insofar as a subnormal divisor does not necessarily have to be normal in the whole group. Every normal divisor is always a subnormal divisor.

example

The subgroup of the symmetrical group created by a reflection is a normal divisor of the Klein group of four , which in turn is normal in . is therefore a subnormal divisor of , but not a normal divisor because in is not . ${\displaystyle Z=\{e,(12)(34)\}}$ ${\displaystyle S_{4}}$ ${\displaystyle V}$ ${\displaystyle S_{4}}$ ${\displaystyle Z}$ ${\displaystyle S_{4}}$ ${\displaystyle ((12)(34))^{(123)}=(13)(24)}$ ${\displaystyle Z}$

literature

• Helmut Wielandt: A generalization of the invariant subgroups. In: Mathematische Zeitschrift 45 (1939), pp. 209–244.