Characteristically simple group

In mathematics , especially in group theory , a group is called characteristically simple if it contains no other characteristic subgroups apart from itself and the trivial subgroup . Some authors also claim that a characteristically simple group should by definition not be a single element, but we do not follow that in this article.

Examples and characteristics

• Every simple group is characteristically simple.

This follows easily from the fact that characteristic subgroups are normal subgroups . We shall see later that characteristically simple groups are not necessarily simple, Klein's group of four is an example.

• Every minimal normal divisor of a group is characteristically simple.

This follows easily from the fact that a characteristic subgroup of a normal subgroup of a group is normal subgroup in that group.

• Let G a characteristically simple group and H a minimal normal subgroup in G . Then, it shows that G the direct sum is a (finite or infinite) family of simple groups, all isomorphic to H are.

For the proof of this statement one needs the lemma of Zorn . If G is finite, the lemma of Zorn can be dispensed with.

• Every characteristically simple group that contains at least one minimal normal subgroup is the direct sum of a (finite or infinite) family of isomorphic, simple groups.

This follows immediately from the previous statement.

• Every finite characteristically simple group is the direct product of a finite number of mutually isomorphic, simple groups.

If G is finite and not trivial, then there is at least one minimal normal sub-divisor, and it is sufficient to apply the previous statement.

• If a finite, resolvable group is characteristically simple, it is elementary Abelian , that is, it is the direct product of finitely many groups that are isomorphic to (see cyclic group ), where p is a prime number . ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$

This follows from the previous statement, because a simple subgroup of a solvable group is Abelian and therefore of prime order . The above conclusion uses Hall's theorem about the existence of certain Hall subgroups in finite, resolvable groups.

• An infinite, characteristically simple group is not necessarily a direct sum of mutually isomorphic, simple groups.

For example, the additive group of rational numbers is characteristically simple (this can be easily shown by the remark that for every rational number q other than 0 defines an automorphism on ), but is not the direct sum of simple subgroups because it has no simple subgroups at all. Indeed, since is Abelian, a simple subgroup would be Abelian and therefore finite, but the only element of finite order is 0. ${\displaystyle \mathbb {Q} }$ ${\displaystyle x\mapsto qx}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$

• One can show that a (finite or infinite) direct sum of isomorphic, simple groups is characteristically simple.

The previous example shows that not every characteristically simple group is of this form.

In particular, Klein's group of four is characteristically simple, which can easily be confirmed directly. This is an example of a characteristically simple group that is not easy. ${\displaystyle \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}}$

Individual evidence

1. ↑ J. Calais: Éléments de théorie des groupes , Paris 1984, p. 257, assumes that G is not one element. WR Scott, Group Theory , Dover Publications 2010, ISBN 0-486-65377-3 , page 73, does not assume this.
2. ^ WR Scott: Group Theory , Dover Publications 2010, ISBN 0-486-65377-3 , page 73, sentence 4.4.2
3. ^ Voir JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, page 106
4. ^ JJ Rotman: An Introduction to the Theory of Groups , 4th edition, Springer-Verlag 1999, p. 109
5. ^ WR Scott: Group Theory , Dover Publications 2010, ISBN 0-486-65377-3 , page 77, exercise 4.4.17