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Characteristically simple group
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This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group
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RANDOM GROUP PROPERTY: Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.
This is a variation of simplicity
Find other variations of simplicity | 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:
- It has no proper nontrivial characteristic subgroup
- The characteristic closure of any nontrivial subgroup is the whole group
- 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:
- For any characteristic subgroup H of G, H is either trivial or the whole group
- 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
- 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 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 | |||
| Strictly simple group | nontrivial, no proper nontrivial ascendant subgroups | click here | ||
| Absolutely simple group | nontrivial, no proper nontrivial serial subgroups | click here | ||
| Additive group of a field | additive group of a field | click here | ||
| 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 | |||
| Group with two conjugacy classes | exactly two conjugacy classes | click here |
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 |
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)
