Normal Subgroups
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Abstract AlgebraSubgroups and Normal SubgroupsNormal Subgroups
Normality Inside Related Subgroups
Overview
Normality is always relative to an ambient group. A subgroup may be normal in one group but not in a larger group, and this is why the notation must mention both and . In this lesson we examine what happens when the surrounding group is replaced by an intermediate subgroup, by a product subgroup, or by a subgroup that intersects a normal subgroup. These results are often used silently later in quotient group arguments, so it is useful to prove them explicitly now.
DEFINITION : Intermediate Subgroup
Let be a group and let be subgroups of . The subgroup is called an \textbf{intermediate subgroup between and } if
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Author
Dr. Bivash Majumder
Assistant Professor in Mathematics
Prabhat Kumar College, Contai