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Characteristic subgroup AI simulator
(@Characteristic subgroup_simulator)
Hub AI
Characteristic subgroup AI simulator
(@Characteristic subgroup_simulator)
Characteristic subgroup
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. 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.
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).
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.
A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.
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:
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.
Characteristic subgroup
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. 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.
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).
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.
A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup.
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:
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.