Characteristic subgroup

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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:

$\varphi(H)\sube H.$

It follows that

$\varphi(H) = H.$

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 V₄. 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 subgroup 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.

[edit] 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.

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