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Descendant subgroup

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Definition

A subgroup H of a group G is termed descendant if we have subgroups H_\alpha of G for every ordinal \alpha such that:

  • H_0 = G
  • H_{\alpha + 1} \ \underline{\triangleleft} \ H_\alpha (i.e., H_{\alpha + 1} is a normal subgroup of H_\alpha) for every ordinal \alpha.
  • If \alpha is a limit ordinal, then H_\alpha = \bigcap_{\gamma < \alpha} H_\gamma.

and such that there is some ordinal \beta such that H_\beta = H.

In terms of the descendant closure operator

The subgroup property of being an descendant subgroup is obtained by applying the descendant closure operator to the subgroup property of being normal.

Relation with other properties

Facts

Descendant-contranormal factorization

This result states that given any subgroup H of G, there is a unique subgroup K containing H such that H is contranormal in K and K is descendant in G.

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes descendance is transitive If H \le K \le G are groups such that H is a descendant subgroup of K and K is a descendant subgroup of G, then H is a descendant subgroup of G.
trim subgroup property Yes Every group is descendant in itself, and the trivial subgroup is descendant in any group.
intermediate subgroup condition Yes descendance satisfies intermediate subgroup condition If H \le K \le G are groups such that H is descendant in G, then H is descendant in K.
strongly intersection-closed subgroup property Yes descendance is strongly intersection-closed If H_i, i \in I, are all descendant subgroups of G, so is the intersection \bigcap_{i \in I} H_i.
image condition No descendance does not satisfy image condition It is possible to have groups G and K, a descendant subgroup H of G and a surjective homomorphism \varphi:G \to K such that \varphi(H) is not a descendant subgroup of K.