Characteristically simple group
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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, invariant under isomorphism
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Definition
Equivalent definitions in tabular format
No. | Shorthand | A nontrivial group is termed characteristically simple if ... | A nontrivial group is characteristically simple if ... |
---|---|---|---|
1 | simple wrt characteristic subgroups | it has no proper nontrivial characteristic subgroup | for any characteristic subgroup of , 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 of , the characteristic closure of (i.e., the subgroup generated by all for ), is the whole group |
3 | characteristic core trivial | the characteristic core of any proper subgroup is trivial | for any proper subgroup of , the characteristic core of (i.e., the intersection of all for ), 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)