This paper mainly researches the relationship among the solution coset of linear equations form the angle of the coset of invariant subgroup,in the course of which the base and the dimension of quotient space have been found out.

Cauchy s theorem and Sylow s theorems for finite groups are proved by means of the double coset decompositions and some basic enumeration techniques.

A Coset Searching Algorithm Based on Quantum Low-density-parity-check Codes;

High-dimensional affine codes were constructed using weak block designs, where the s-dimensional coset of s-dimensional affine geometry G(m,p) over a finite field F_p was taken as an informational bit, and every s-dimensional coset family corresponding each element was regarded as a check line.

With regard to the section of coset in algebraic structure, a series of well-mastered concepts including equivalent relationship and division are applied, and typical examples are cited to introduce the conception of equivalent relationship-coset relationship defined through subgroup.

The CI-property of coset graphs of groups with order qp;

The Cayley graph and Sabidussi coset graph are the classical representativities of these graphs.

The classical representations of these type graphs are Cayleygraphs and Sabidussi Coset graphs.