Characteristic subgroup: Difference between revisions
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=== Finite example === | === Finite example === | ||
Consider the group <math>G = S_3 \times \mathbb{Z}_2</math> (the group of order 12 that is the direct product of the [[symmetric group]] of order 6 and a [[cyclic group]] of order 2). The center of <math>G</math> is isomorphic to its second factor <math>\mathbb{Z}_2</math>. Note that the first factor, <math>S_3</math>, contains subgroups isomorphic to <math>\mathbb{Z}_2</math>, for instance <math>{e, (12)}</math>; let <math>f: \mathbb{Z}_2 \rarr \text{S}_3</math> be the morphism mapping <math>\mathbb{Z}_2</math> onto the indicated subgroup. Then the composition of the projection of <math>G</math> onto its second factor <math>\mathbb{Z}_2</math>, followed by <math>f</math>, followed by the inclusion of <math>S_3</math> into <math>G</math> as its first factor, provides an endomorphism of <math>G</math> under which the image of the center, <math>\mathbb{Z}_2</math>, is not contained in the center, so here the center is not a fully characteristic subgroup of <math>G</math>. | |||
Consider the group | |||
=== Cyclic groups === | === Cyclic groups === | ||
Latest revision as of 22:53, 19 May 2026
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group.[1][2] Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group.
Definition
A subgroup H of a group G is called a characteristic subgroup if for every automorphism φ of G, one has φ(H) ≤ H; then write H char G.
It would be equivalent to require the stronger condition φ(H) = H for every automorphism φ of G, because φ−1(H) ≤ H implies the reverse inclusion H ≤ φ(H).
Basic properties
Given H char G, every automorphism of G induces an automorphism of the quotient group G/H, which yields a homomorphism Aut(G) → Aut(G/H).
If G has a unique subgroup H of a given index, then H is characteristic in G.
Related concepts
Normal subgroup
A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.
- ∀φ ∈ Inn(G)： φ(H) ≤ H
Since Inn(G) ⊆ Aut(G) and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic. Here are several examples:
- Let H be a nontrivial group, and let G be the direct product, H × H. Then the subgroups, {1} × H and H × {1}, are both normal, but neither is characteristic. In particular, neither of these subgroups is invariant under the automorphism, (x, y) → (y, x), that switches the two factors.
- For a concrete example of this, let V be the Klein four-group (which is isomorphic to the direct product, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2 \times \mathbb{Z}_2} ). Since this group is abelian, every subgroup is normal; but every permutation of the 3 non-identity elements is an automorphism of V, so the 3 subgroups of order 2 are not characteristic. Here V = {e, a, b, ab} . Consider H = {e, a} and consider the automorphism, T(e) = e, T(a) = b, T(b) = a, T(ab) = ab; then T(H) is not contained in H.
- In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, {1, −1}, is characteristic, since it is the only subgroup of order 2.
- If n > 2 is even, the dihedral group of order 2n has 3 subgroups of index 2, all of which are normal. One of these is the cyclic subgroup, which is characteristic. The other two subgroups are dihedral; these are permuted by an outer automorphism of the parent group, and are therefore not characteristic.
Strictly characteristic subgroup
A strictly characteristic subgroup, or a distinguished subgroup, is one which is invariant under surjective endomorphisms. For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being strictly characteristic is equivalent to characteristic. This is not the case anymore for infinite groups.
Fully characteristic subgroup
For an even stronger constraint, a fully characteristic subgroup (also, fully invariant subgroup) of a group G, is a subgroup H ≤ G that is invariant under every endomorphism of G (and not just every automorphism):
- ∀φ ∈ End(G)： φ(H) ≤ H.
Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. The commutator subgroup of a group is always a fully characteristic subgroup.[3][4]
Every endomorphism of G induces an endomorphism of G/H, which yields a map End(G) → End(G/H).
Verbal subgroup
An even stronger constraint is verbal subgroup, which is the image of a fully invariant subgroup of a free group under a homomorphism. More generally, any verbal subgroup is always fully characteristic. For any reduced free group, and, in particular, for any free group, the converse also holds: every fully characteristic subgroup is verbal.
Transitivity
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.
- H char K char G ⇒ H char G.
Moreover, while normality is not transitive, it is true that every characteristic subgroup of a normal subgroup is normal.
- H char K ⊲ G ⇒ H ⊲ G
Similarly, while being strictly characteristic (distinguished) is not transitive, it is true that every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic.
However, unlike normality, if H char G and K is a subgroup of G containing H, then in general H is not necessarily characteristic in K.
- H char G, H < K < G ⇏ H char K
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, Sym(3) × Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} / 2 \mathbb{Z}} , has a homomorphism taking (π, y) to ((1, 2)y, 0), which takes the center, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \times \mathbb{Z} / 2 \mathbb{Z}} , into a subgroup of 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
Examples
Finite example
Consider the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = S_3 \times \mathbb{Z}_2} (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is isomorphic to its second factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2} . Note that the first factor, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_3} , contains subgroups isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2} , for instance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {e, (12)}} ; let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \mathbb{Z}_2 \rarr \text{S}_3} be the morphism mapping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2} onto the indicated subgroup. Then the composition of the projection of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} onto its second factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2} , followed by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , followed by the inclusion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_3} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} as its first factor, provides an endomorphism of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} under which the image of the center, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2} , is not contained in the center, so here the center is not a fully characteristic subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} .
Cyclic groups
Every subgroup of a cyclic group is characteristic.
Subgroup functors
The derived subgroup (or commutator subgroup) of a group is a verbal subgroup. The torsion subgroup of an abelian group is a fully invariant subgroup.
Topological groups
The identity component of a topological group is always a characteristic subgroup.
See also
References
- ↑ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
- ↑ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
- ↑ Scott, W.R. (1987). Group Theory. Dover. pp. 45–46. ISBN 0-486-65377-3.
- ↑ Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (2004). Combinatorial Group Theory. Dover. pp. 74–85. ISBN 0-486-43830-9.