Editing Characteristic subgroup (section)

== Containments ==
Every subgroup that is fully characteristic is certainly strictly characteristic and characteristic; but a characteristic or even strictly characteristic subgroup need not be fully characteristic.

The [[center of a group]] is always a strictly characteristic subgroup, but it is not always fully characteristic.  For example, the finite group of order 12, {{math|Sym(3) × <math>\mathbb{Z} / 2 \mathbb{Z}</math>}}, has a homomorphism taking {{math|(''π'', ''y'')}} to {{math|((1, 2){{sup|''y''}}, 0)}}, which takes the center, <math>1 \times \mathbb{Z} / 2 \mathbb{Z}</math>, into a subgroup of {{math|Sym(3) × 1}}, which meets the center only in the identity.

The relationship amongst these subgroup properties can be expressed as:
:[[Subgroup]] ⇐ [[Normal subgroup]] ⇐ '''Characteristic subgroup''' ⇐ Strictly characteristic subgroup ⇐ [[Fully characteristic subgroup]] ⇐ [[Verbal subgroup]]