Fully invariant subgroup

Fully invariant subgroups are considered subgroups in the mathematical subfield of group theory with an additional property. This additional property means that the subgroup is invariant under each endomorphism of the group.

definition

It is a group . A subgroup is called fully invariant , if ${\displaystyle G}$ ${\displaystyle U\subset G}$

${\displaystyle \varphi (U)\subset U}$   for all endomorphisms of the group . ${\displaystyle \varphi }$ ${\displaystyle G}$

Examples

• Obviously the trivial subgroup and the group itself are always fully invariant subgroups. If these are the only fully invariant subgroups, the group is called fully invariant-simple. ${\displaystyle \{1\}}$
• Since homomorphic images of commutators are commutators again, the subgroups formed with them result in fully invariant groups when defining resolvable or nilpotent groups:

${\displaystyle G^{(0)}:=G}$and for , the so-called series of derived subgroups , it consists of fully invariant subgroups. ${\displaystyle G^{(n)}:=[G^{(n-1)},G^{(n-1)}]}$ ${\displaystyle n>0}$

${\displaystyle \gamma _{1}G:=G}$and for , the so-called descending central series, it consists of fully invariant subgroups. ${\displaystyle \gamma _{n}G:=[G^{(n-1)},G]}$ ${\displaystyle n>0}$

• If the subgroup generated by all -th powers denotes , they are also fully invariant. ${\displaystyle G^{n}}$ ${\displaystyle n}$ ${\displaystyle G^{n}}$
• If a group has exactly one - Sylow group for a prime number , this is fully invariant. This is always the case in Abelian groups. ${\displaystyle p}$ ${\displaystyle p}$
• Centers of groups are generally not fully invariant. For example, the center of A ₄ ℤ _{2 is} not fully invariant. ${\displaystyle \times }$

Remarks

Obviously, the following implications exist for subgroups of a group:

fully invariant   characteristic normal   subgroup ${\displaystyle \Rightarrow }$     ${\displaystyle \Rightarrow }$     ${\displaystyle \Rightarrow }$

The inversions do not apply. For example, centers of groups are always characteristic, but generally not fully invariant, as can be seen from the above examples.

In group theory, one often considers groups with an operator range. This is a set , so that an endomorphism of the group is defined for each . For example, you can - modules as Abelian groups consider, so that at any ring member of the linear operator of scalar multiplication with is explained in this case . Or you can equip each group with the operator area , whereby the conjugation with is for one . Then one is interested in so-called -substructures that respect these operators, i.e. remain invariant under the endomorphisms of the operator range. The isomorphism theorems or the Jordan-Hölder theorem also apply in this context . It is clear that fully invariant subgroups are always -substructures. ${\displaystyle \Omega }$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle \varphi _{\omega }}$ ${\displaystyle R}$ ${\displaystyle r\in R}$ ${\displaystyle \varphi _{r}}$ ${\displaystyle r}$ ${\displaystyle \Omega =R}$ ${\displaystyle G}$ ${\displaystyle \Omega =G}$ ${\displaystyle \varphi _{g}}$ ${\displaystyle g\in G}$ ${\displaystyle g\in G}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

The fully invariant subgroups of a group form a closed association . Fully invariance is also transitive, that is, if a fully invariant subgroup of and a fully invariant subgroup in , then is also fully invariant in . ${\displaystyle U\subset G}$ ${\displaystyle G}$ ${\displaystyle V\subset U}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle G}$

Individual evidence

1. ^ Wilhelm Specht : Group theory. Springer-Verlag (1956), Chapter 1.3.4: Chakateristic and fully invariant subgroups , definition 7.
2. ^ DJS Robinson : A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , Chapter 1.5: Characteristic and Fully-Invariant Subgroups.
3. ^ Wilhelm Specht : Group theory. Springer-Verlag (1956), chapter 1.3.4. according to definition 7.
4. ^ DJS Robinson: A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , Chapter 5.1: The Lower and Upper Central Series.
5. ^ Thomas W. Hungerford: Algebra. Springer Verlag (2003), ISBN 978-0-387-90518-1 , Lemma 7.13 (ii)
6. ^ DJS Robinson: A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , exercise 1.5.9.
7. ^ Thomas W. Hungerford: Algebra. Springer Verlag (2003), ISBN 978-0-387-90518-1 , page 103.
8. ^ Wilhelm Specht: Group theory. Springer-Verlag (1956), Chapter 1.3.4: Sentence 27 and 27 *