Characteristicity satisfies partition difference condition
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., partition difference condition)
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Statement
Suppose is a group and
is a subgroup of
such that
has a partition into subgroups
, with
containing at least two elements. Then, if all except possibly one of the
is a characteristic subgroup of
, all the
s are characteristic subgroups of
.