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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 1.3 Equivalence of definitions 2 Formalisms 2.1 In terms of the simple group operator 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties 4 Metaproperties 4.1 Direct products 5 References 5.1 Textbook references

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a group property: a property that can be evaluated to true/false for any given group
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This is a variation of simplicity
View a complete list of variations of simplicity OR read a survey article on varying simplicity

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 σ(H) for ), is the whole group G
3. For any proper subgroup H of G, the characteristic core of H (i.e., the intersection of all σ(H) for ), 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 all 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

• Simple group
• Strictly simple group
• Absolutely simple group
• Additive group of a field
• Group of p-adic integers
• Group whose automorphism group is transitive on non-identity elements: For full proof, refer: Automorphism group is transitive on non-identity elements implies characteristically simple

Weaker properties

• FC-simple group

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)