Subgroup series

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]

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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 $<S_v,S_v'>$.

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 $V$.