B tech. Discrete mathematics (I. T & Comp. Science Engg.) Syllabus

COSET AND NORMAL SUBEROUP:

Left Coset : Let (H, ) be a subgroup of (G, ). For any a  G, the set of aH defined by aH  {a  h / h  H} is called the left coset of H in G

determined by the element aG. The element a is called the representative element of the left coset aH.
Right Coset : Let (H, ) be a subgroup of (G, ). For any a  G, the set of Ha defined by

^{Ha } ^{h * a | h H}

is called the right coset of H in G determined by the element aG. The element a is called the representative element of the right coset Ha.

Theorem : Let (H,  ) be a subgroup of (G,  ). The set of left cosets of H in G form a partition of G. Every element of G belongs to one and only one left coset of H in G.