Characteristically simple group

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Definition

Symbol-free definition

A nontrivial group is said to be characteristically simple if it satisfies the following equivalent conditions:

1. It has no proper nontrivial characteristic subgroup
2. The characteristic closure of any nontrivial subgroup is the whole group
3. The characteristic core of any proper subgroup is trivial

When the group is finite, this is equivalent to it being a direct product of pairwise isomorphic simple groups.

Definition with symbols

A nontrivial group $G$ is termed characteristically simple if it satisfies the following equivalent conditions:

1. For any characteristic subgroup $H$ of $G$, $H$ is either trivial or the whole group
2. For any nontrivial subgroup $H$ of $G$, the characteristic closure of $H$ (i.e., the subgroup generated by all $\sigma (H)$ for $\sigma \in \mathrm{A}\mathrm{u}\mathrm{t}(G)$), is the whole group $G$
3. For any proper subgroup $H$ of $G$, the characteristic core of $H$ (i.e., the intersection of all $\sigma (H)$ for $\sigma \in \mathrm{A}\mathrm{u}\mathrm{t}(G)$), is the trivial subgroup (i.e., just the identity element)

When the group $G$ is finite, this is equivalent to $G$ being a direct product of pairwise isomorphic simple groups.

Equivalence of definitions

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

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			|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			|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)