Characteristically simple group

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Definition

Equivalent definitions in tabular format

No.	Shorthand	A nontrivial group is termed characteristically simple if ...	A nontrivial group $G$ is characteristically simple if ...
1	simple wrt characteristic subgroups	it has no proper nontrivial characteristic subgroup	for any characteristic subgroup $H$ of $G$, $H$ is either trivial or the whole group
2	characteristic closure whole group	the characteristic closure of any nontrivial subgroup is the whole group	or any nontrivial subgroup $H$ of $G$, the characteristic closure of $H$ (i.e., the subgroup generated by all $\sigma(H)$ for $\sigma \in \operatorname{Aut}(G)$), is the whole group $G$
3	characteristic core trivial	the characteristic core of any proper subgroup is trivial	for any proper subgroup $H$ of $G$, the characteristic core of $H$ (i.e., the intersection of all $\sigma(H)$ for $\sigma \in \operatorname{Aut}(G)$), is the trivial subgroup (i.e., just the identity element)

When the group is finite, this is equivalent to it being a direct product of pairwise isomorphic simple groups; however, this is not true for infinite characteristicaly simple groups.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of characteristically simple group

Examples

• In the finite case, the characteristically simple groups are precisely the direct powers of simple groups. Thus, the characteristically simple abelian groups are precisely the elementary abelian groups, and the characteristically simple non-abelian groups are precisely the direct powers of the simple non-abelian groups.
• In the infinite case, there are more possibilities. For instance, the additive group of any vector space over any field is characteristically simple and abelian. On the non-abelian side, there are examples such as McLain's group over a field.

Formalisms

In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: characteristic subgroup
View other properties obtained by applying the simple group operator

The group property of being characteristically simple is obtained by applying the simple group operator to the trim subgroup property of being characteristic.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Simple group	nontrivial, no proper nontrivial normal subgroups			|FULL LIST, MORE INFO
Strictly simple group	nontrivial, no proper nontrivial ascendant subgroups			Simple group|FULL LIST, MORE INFO
Absolutely simple group	nontrivial, no proper nontrivial serial subgroups			Simple group|FULL LIST, MORE INFO
Additive group of a field	additive group of a field			Group whose automorphism group is transitive on non-identity elements|FULL LIST, MORE INFO
Group of p-adic integers
Group whose automorphism group is transitive on non-identity elements		Automorphism group is transitive on non-identity elements implies characteristically simple		|FULL LIST, MORE INFO
Group with two conjugacy classes	exactly two conjugacy classes			Group whose automorphism group is transitive on non-identity elements|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group that is the characteristic closure of a singleton subset				|FULL LIST, MORE INFO

Metaproperties

Direct products

The direct product of two characteristically simple groups is characteristically simple if and only if they are powers of the same simple group. Note that the simple group is unique upto isomorphism.

References

Textbook references

• Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754, ^{More info}, Page 25
• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 16 (definition in paragraph, preceding Theorem 1.4)