CHARACTERISTIC SUBGROUP

In mathematics, a 'characteristic subgroup' of a group ''G'' is a subgroup ''H'' that is invariant under each automorphism of ''G''. That is, if φ : ''G'' → ''G'' is a group automorphism (a bijective homomorphism from the group ''G'' to itself), then for every ''x'' in ''H'' we have φ(''x'') ∈ ''H'':
:
It follows that
:
In symbols, one denotes the fact that ''H'' is a characteristic subgroup of ''G'' by
:$H,mathrm\{char\},G.$
In particular, characteristic subgroups are invariant under inner automorphisms, so they are normal subgroups. However, the converse is not true; for example, consider the Klein group ''V''₄. Every subgroup of this group is normal; but all 6 permutations of the 3 non-identity elements are automorphisms, so the 3 subgroups of order 2 are not characteristic.
On the other hand, if ''H'' is a normal subgroup of ''G'', and there are no other subgroups of the same order, then ''H'' must be characteristic; since automorphisms are order-preserving.
A related concept is that of a 'distinguished subgroup'. In this case the subgroup ''H'' is invariant under the applications of surjective endomorphisms. For a finite group this is the same, because surjectivity implies injectivity, but not for an infinite group: a surjective endomorphism is not necessarily an automorphism.
For an even stronger constraint, a 'fully characteristic subgroup' (also called a 'fully invariant subgroup') ''H'' of a group ''G'' is a group remaining invariant under every endomorphism of ''G''; in other words, if ''f'' : ''G'' → ''G'' is any homomorphism, then ''f''(''H'') is a subgroup of ''H''.
Every subgroup that is fully characteristic is certainly distinguished and therefore characteristic; but a characteristic or even distinguished subgroup need not be fully characteristic. The center of a group is easily seen to always be a distinguished subgroup, but it is not always fully characteristic.

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Example

Example

See also

Example

Consider the group ''G'' = S₃ × Z₂ (the group of order 12 which is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of ''G'' is its second factor Z₂. Note that the first factor S₃ contains subgroups isomorphic to Z₂, for instance {identity,(12)}; let ''f'': Z₂ → S₃ be the morphism mapping Z₂ onto the indicated subgroup. Then the composition of the projection of ''G'' onto its second factor Z₂, followed by ''f'', followed by the inclusion of S₃ into ''G'' as its first factor, provides an endomorphism of ''G'' under which the image of the center Z₂ is not contained in the center, so here the center is not a fully characteristic subgroup of ''G''.
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The derived subgroup (or commutator subgroup) of a group is always a fully characteristic subgroup, as is the torsion subgroup of an abelian group.
The property of being characteristic or fully characteristic is transitive; if ''H'' is a (fully) characteristic subgroup of ''K'', and ''K'' is a (fully) characteristic subgroup of ''G'', then ''H'' is a (fully) characteristic subgroup of ''G''.
Moreover, while it is not true that every normal subgroup of a normal subgroup is normal, it is true that every characteristic subgroup of a normal subgroup is normal. Similarly, while it is not true that every distinguished subgroup of a distinguished subgroup is distinguished, it is true that every fully characteristic subgroup of a distinguished subgroup is distinguished.
The relationship amongst these subgroup properties can be expressed as:
:subgroup ← normal subgroup ← characteristic subgroup ← distinguished subgroup ← fully characteristic subgroup

Example

Every subgroup of a cyclic group is characteristic.

See also

★ characteristically simple group.