Properties of Cosets

Prove the two properties: 1) If $aH = bH$, then $a \in bH$. 2) If $a \in bH$, then $aH = bH$.

The problem at hand is a fundamental exploration within the study of group theory, specifically regarding cosets. Cosets arise naturally when you partition a group into disjoint subsets using a subgroup, offering deep insights into the structure and properties of groups. The two properties in question are essential in understanding the equivalence conditions within a group and how these relate to cosets.

The first property states that if two left cosets of a group are equal, then you can find one element of the group in the other coset. This principle underlies the nature of group actions and the symmetry within the set of cosets. Understanding this highlights the concept of equivalence classes, as each element in a group corresponds to a distinct coset.

The second property is the converse of the first, establishing that if an element belongs to the coset of another, then their cosets are indeed the same. This symmetric relation between elements and their respective cosets encapsulates a vital component of group theory. It not only emphasizes the bijective nature of this relationship but also ties closely to Lagrange's Theorem, which can be used to deduce important characteristics of the group structure such as the order of subgroups and the group itself. Together, these properties are foundational, offering a glimpse into more complex concepts such as normal subgroups and quotient groups.