Minimal normal subgroup with order greater than index is characteristic

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Statement

For an arbitrary group

A Minimal normal subgroup (?) of a group, such that the order of the subgroup is greater than its index, must be a Characteristic subgroup (?).

For a finite group

A minimal normal subgroup of a finite group, whose order is more than the squareroot of the order of the group, is characteristic.

Related facts

  1. Minimal normal subgroup with order not dividing index is characteristic
  2. Simple normal subgroup with order not dividing index is fully characteristic