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Encyclopedia > 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 : GG is a group automorphism (a bijective homomorphism from the group G to itself), then for every x in H we have φ(x) ∈ H: Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

It follows that

In symbols, one denotes the fact that H is a characteristic subgroup of G by

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. In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, the Klein four-group (or just Klein group or Viergruppe, often symbolized by the letter V), named after Felix Klein, is a group with four elements, the smallest non-cyclic group. ...

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. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ...

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 : GG 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. In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z &#8712; G | gz = zg for all g &#8712; G} Note that Z(G) is a subgroup of...

### 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. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

The derived subgroup (or commutator subgroup) of a group is always a fully characteristic subgroup, as is the torsion subgroup of an abelian group. In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>. Note that the set of all commutators of the group is, generally, not a group (in... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

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. In mathematics, the word transitive admits at least two distinct meanings: A group G acts transitively on a set S if for any x, y &#8712; S, there is some g &#8712; G such that gx = y. ...

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 Results from FactBites:

 PlanetMath: characteristic subgroup (187 words) This is precisely the definition of a normal subgroup. This is version 10 of characteristic subgroup, born on 2002-07-21, modified 2006-02-17. For the phrase, "H is a characteristic subgroup of G", the only notation I have ever seen in books and used by professors is "H char G".
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