# Left-complemented subset

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## Contents

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## Definition

### Symbol-free definition

A subset of a group is termed a left complemented subset if it satisfies the following equivalent conditions:

• There exists another subset such that every element of the group can be written as the product of an element from the other subset and an element of the original subset, in a unique way.
• There is a system of translates of the subset (under left multiplication by the group) that form a partition of the group.

We can analogously define right complemented subset.

### Definition with symbols

A subset $T$ of a group $G$ is termed a left complemented subset if it satisfies the following equivalent conditions:

• There exists another subset $S$ such that every element of $G$ can be uniquely written in the form $st$ where $s$ is in $S$ and $t$ is in $T$.
• There exists a collection of translates $xT$ of $T$ such that the $xT$ are pairwise disjoint and their union is the whole of $G$.