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 $V$.