Characteristicity is transitive
From Groupprops
DIRECT: The fact or result stated in this article has a trivial/direct/straightforward proof provided we use the correct definitions of the terms involved
View other results with direct proofs
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
|
Property "Page" (as page type) with input value "{{{property}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Property "Page" (as page type) with input value "{{{metaproperty}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Statement
Property-theoretic statement
The subgroup property of being characteristic satisfies the subgroup metaproperty of being transitive.
Verbal statement
A characteristic subgroup of a characteristic subgroup is characteristic in the whole group.
Symbolic statement
Let 
be a characteristic subgroup of 
, and a characteristic subgroup of 
. Then, is a characteristic subgroup of .
Proof
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]References
Textbook references
- Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info, Page 17, Lemma 4
- A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info, Page 28, Characteristic and Fully invariant subgroups, 1.5.6(ii)
