Subgroup series

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Definition

Symbol-free definition

A subgroup series in a group is an order-preserving map from a totally ordered indexing set to the collection of subgroups of the group, ordered by inclusion.

Definition with symbols

A subgroup series in a group G is a map from a totally ordered indexing set W to the collection of subgroups of G, that sends w in W to H_w, such that if v < w, then H_v is a subgroup of H_w.

Particular cases

Well-ordered indexing set (for minimum)

When the indexing set is well-ordered, the subgroup series is termed an ascending series. For any well-ordered indexing set, every element is either a successor element or a limit element. If, further, we have the property that for a limit element, the corresponding subgroup is the union of the subgroups below it, we call the ascending series limit-tight.

Well-ordered indexing set (for maximum)

When every subset has a maximum (that is, the indexing set is well-ordered with respect to maximum), the subgroup series is termed a descending series. If, further, we have the property that for a limit element, the corresponding subgroup is the intersection of the subgroups above it, we call the descending series limit-tight.

Operations on subgroup series

Concatenation of subgroup series

Consider a subgroup series S_1 in a group G where the intersection of all members of the series is H. Let S_2 be a subgroup series in H. Then the concatenation of S_2 with S_1 is a subgroup series of G which essentialyl concatenates the subgroup series. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Intersection of subgroup series

Let S and S' be two subgroup series in a group G both indexed by the same indexing set V. Then, the intersection ofS and S', denoted S \cap S', associated to each element v \in V the subgroup S_v \cap S_v'.

Join of subgroup series

Let S and S' be two subgroup series in a group G both indexed by the same indexing set V. Then, the intersection ofS and S', denoted S \cap S', associated to each element v \in V the subgroup <S_v,S_v'>.

Maps from subgroup properties to subgroup series properties

The successor condition map

Given a subgroup property p, the successor condition map gives a subgroup series property that a subgroup series satisfies if and only if:

  • For any member H_w, H_w satisfies p as a subgroup of the intersection of all H_v for v > w.
  • For any member H_w, the subgroup generated by H_v for v < w satisfies the property p as a subgroup of H_w.

The comparison condition map

Given a subgroup property p, the comparison condition map gives the property of being a subgroup series where, given any subsets U and V of W such that every element of U is less than every element of V the subgroup generated by all the subgroups corresponding to U satisfies p as a subgroup of the intersection of all the subgroups corresponding to V.