Definition:Left Coset/Also defined as
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Left Coset: Also defined as
It is usual for the algebraic structure $S$ in fact to be a group.
Hence, when $\struct {S, \circ}$ is a group, the left coset of $H$ by $x$ is the equivalence class of $x$ defined by left congruence modulo $H$.
Some sources (see P.M. Cohn: Algebra Volume 1 (2nd ed.), for example) order the operands in the opposite direction, and hence $H \circ x$ is a left coset.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.15$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions