Normal Subgroups
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Abstract AlgebraSubgroups and Normal SubgroupsNormal Subgroups
Definition and First Examples of Normal Subgroups
Overview
Having learned how cosets partition a group and how indices measure the size of that partition, the next structural question is whether cosets can themselves be multiplied without ambiguity. This question leads naturally to normal subgroups. A subgroup may sit inside a group in many ways, but a normal subgroup sits symmetrically: its left cosets and right cosets coincide. Normal subgroups are the subgroups from which quotient groups are built, so they form one of the central bridges from elementary group theory to homomorphisms and isomorphism theorems. Students often think that every subgroup behaves well with cosets; the first purpose of this lesson is to separate ordinary subgroup behaviour from normal subgroup behaviour.
DEFINITION : Normal Subgroup
Let be a group and let be a subgroup of . Then is called a \textbf{normal subgroup} of if
It is denoted by
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Author
Dr. Bivash Majumder
Assistant Professor in Mathematics
Prabhat Kumar College, Contai