Left and Right Cosets

Abstract Algebra · Axiom Academy

Understanding how subgroups create partitions through cosets, and why multiplication order matters in non-abelian groups. 1. Defining Left and Right Cosets In words: a left coset consists of all products where runs through all elements of the subgroup. A right coset reverses the multiplication order. Let's see how to build a coset step by step. Consider a concrete example: with subgroup . Example: Z₆ with subgroup H = 0, 3 Similarly, 2 + H = 2, 5 and the pattern continues. This happens because for any two elements , either or . A remarkable fact: every coset of has exactly elements, the same as the subgroup itself. 5. When Left ≠ Right: Matrix Groups In abelian groups, left and right cosets coincide. But in non-abelian groups, they can differ dramatically. Let's see a matrix group example. Matrix Example: Upper Triangular Matrices Let be 2×2 invertible matrices and be diagonal matrices. Consider: The left coset gH contains all matrices with first row (1, 1), while the right coset Hg has different structure depending on how the diagonal acts.

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