15.2.5. Cosets

Imagine you are given a group and a subgroup for the group. Again, they may be finite, infinite and so on because the definition does not put any restriction. And imagine you are given a group element g from the bigger group. The cosets are defined with respect to your subgroups. But they are defined with respect to elements which are chosen from the bigger group.

In group theory, cosets are created by taking a group element and a subgroup. There are two types of cosets: left and right. The left coset of a subgroup H is formed by taking an element g from the group G and multiplying it by every element in H, denoted as gH. The right coset is formed by multiplying every element in H by the element g, denoted as Hg. The important point is that these cosets can be created regardless of whether the element g belongs to the subgroup H or not.

Think of a classroom (the group G) and a specific group of students (the subgroup H) that are part of a larger study group. If you call out one specific student from the classroom (the element g), then calling out the names of the students in their study group as they relate to that one student is similar to forming a left coset. Doing the reverse, where each student calls out their study partner in relation to the original student, represents forming a right coset.

The left coset of the subgroup $H$ is denoted by $gH$. It is basically the collection of all group elements which I obtained by performing the group operation on the chosen group element $g$ with all elements of the subgroup $H$. In contrast, the right coset is defined similarly, but now the elements of the subgroup $H$ will be operated with $g$. That will be your right coset, denoted by $Hg$.

When defining cosets, we distinguish between left and right based on who is the first operand in the operation. In the left coset $gH$, the group element g is applied to the starting side of all the subgroup elements, while in the right coset $Hg$, the subgroup elements are applied to the starting side of the element g. This distinction is crucial as it affects the resultant group elements formed in each case. Notably, changing the element g will typically yield different cosets.

Imagine a bakery with various types of bread (the group) and a specific type of bread (the subgroup). If you take one specific type of bread g and find out the taste combinations by adding different spreads (the left coset), that determines how g can be paired with those spreads. If instead, you start with the spreads and see how they can be applied to g (the right coset), you're observing exactly how the spreads enhance or change g's taste, leading to different flavor combinations.

The first thing that we can prove here is the following. If H is a finite subgroup then you take any element from the bigger group, the corresponding left coset and right coset have the same cardinality as the cardinality of your finite subgroup.

Every element in the finite subgroup H will generate a unique element in the left or right coset when combined with the group element g, maintaining consistency. This means the number of elements (cardinality) in the left coset $gH$ or in the right coset $Hg$ will equal the number of elements in H because we can't have duplicate results due to the unique nature of group operations. For example, if one element leads to the same result as another in generating cosets, it would mean the respective subgroup members must be identical.

If you think about a dance group (the subgroup) that has several dancers, each performing different styles (group operations), then calling out the style of one dancer (the group element g) will produce a unique combination of all styles paired with that particular dancer. No two unique dancer styles can produce the same output when combined with g, demonstrating that the combination's count remains constant.

If you are given any group and its subgroup, then you take any two elements from the parent group, say g1 and g2, then the left cosets of H with respect to g1 and g2 will either be the same or completely disjoint.

This property states that any two left cosets related to a common subgroup either completely overlap (are identical) or have no elements in common at all (are disjoint). This characteristic arises from the nature of groups and the associative operations applied; hence no partial overlaps happen. This is fundamentally important because it shapes our understanding of how groups can be partitioned.

Imagine you are organizing a sports tournament. Each team (element from the group) must either completely share their players (same left coset) or have completely different players (disjoint left coset). One cannot have a situation where part of one team shares players with another while parts don’t because that would break the integrity of the team's uniqueness.

We will prove that this is a general result. So, what we are going to prove here is that any two left cosets of a subgroup will be either exactly the same or completely disjoint.

To show cosets behave as equivalence classes, we leverage the idea of an equivalence relation. By defining a relation where two elements belong to the same equivalence class if they are part of the same coset, we can demonstrate that this relation satisfies reflexivity, symmetry, and transitivity. Thus, this provides a structural understanding that left cosets partition the entire group, making them key to our group theory interpretations.

Think of attendees at a wedding, where each table represents a coset of seating arrangements. Each person welcomed (group element) can either sit at one table (one equality class) or at another table with no overlap, creating a distinct group per table (disjoint). If someone calls out a new table arrangement (the subgroup's position), it’s like establishing a new equivalence class throughout the break.

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