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Characteristic subgroup
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== Transitivity == The property of being characteristic or fully characteristic is [[transitive relation|transitive]]; if {{math|''H''}} is a (fully) characteristic subgroup of {{math|''K''}}, and {{math|''K''}} is a (fully) characteristic subgroup of {{math|''G''}}, then {{math|''H''}} is a (fully) characteristic subgroup of {{math|''G''}}. :{{math|''H'' char ''K'' char ''G'' β ''H'' char ''G''}}. Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal. :{{math|''H'' char ''K'' β² ''G'' β ''H'' β² ''G''}} Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic. However, unlike normality, if {{math|''H'' char ''G''}} and {{math|''K''}} is a subgroup of {{math|''G''}} containing {{math|''H''}}, then in general {{math|''H''}} is not necessarily characteristic in {{math|''K''}}. :{{math|''H'' char ''G'', ''H'' < ''K'' < ''G'' β ''H'' char ''K''}}
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