Understanding Cosets and Congruence Modulo Subgroups in Mathematics
Explore the concept of cosets within groups, defined as the set of all possible products between elements of a subgroup and the group's elements. Learn about right and left cosets, congruence modulo a subgroup, and the properties of equivalence relations within group theory. Dive into examples and theorems to deepen your understanding of algebraic structures in mathematics.
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DEPARTMENT OF MATHEMATICS SGGSJ GOVERNMENT COLLEGE PAONTA SAHIB
COURSE NAME : ALGEBRA COURSE CODE : 202
CONTENTS COSETS (DEFINITION) AN EXAMPLE BASED ON COSETS RIGHT AND LEFT CONGRUENCE MODULO A SUBGROUP Definition. Theorem.
COSETS Definition: Let H be a subgroup of a group G. If a G, then the set Ha={ha: h H} is called a right coset of H in G deter,imed by a and the set aH={ ha:h H} is called the left coset of H in G determined by a. Note: When G is an abelian group then there is no distinction between a left coset and a right coset i.e., left coset = right coset i.e., aH = Ha
AN EXAMPLE BASED ON COSETS EXAMPLE: Find the right cosets of the subgroups {1,-1} of athe group {1, -1, i, -i} under multiplication. SOLUTION. G = {1, -1, i, -i} is a group under multiplication. H = {1, -1} is a subgroup of G. The right cosets of H in G are H1, H(-1), Hi, H(-i) H.1={1(1), -1(1)}={1, -1}=H. H(-1)={1(-1), -1(-1)}={-1,1}= H. Hi={1(i), -1(i)} = {i ,-i}={i, -i}. H(-i)={1(-1), -1(-i)} = {-i, i}.
RIGHT AND LEFT CONGRUENCE MODULO A SUBGROUP Let H be a subgroup of a groups G. Then a relation R in G as follow: a R b iff ab-1 H. This relation R is denoted by the symbol r mod H i.e., a rb mod H(read as, a is right congruent to bmodulo H ) or a b (mod H), iff ab-1 H.
Similarly, when the relation R is defined as follow: a R b iff a-1b H. where R is denoted by the symbol lmod H i.e., a lb mod H(read as, a is left congruent to b modulo H ) or a b (mod H), iff a-1b H.
THEORM Let H be a subgroup pf a group G. Then the relation a b mod H is an equivalence relation. Proof. Reflexivity:Let a be any element of G. Then aa-1 =e G, since H is a subgroup of G. Therefore a a mod H for all a G. Hence the relation is reflexive. Symmetry: Let a b mod H ab-1 H. (ab-1)-1 = ba-1 H [ H is a subgroup] b amod H. This show that the relation is symmetric.
Transitivity:Let ab mod H and bc mod H ab-1 H and bc-1 H (ab-1)(bc-1 ) H [ H is a subgroup] a(b-1b)c-1 H a(e)c-1 H ac-1 H a c mod H. Hence the relation is transitive also. Therefore the relation (congruence mod H) i.e., mod H is an equivalence relation.
