Subgroup series

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 $$H_w$$.

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.

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.

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 of$$S$$ 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 of$$S$$ and $$S'$$, denoted $$S \cap S'$$, associated to each element $$v \in V$$ the subgroup $$$$.

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 $$V$$.