Applications of Lagrange Theorem
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Abstract AlgebraSubgroups and Normal SubgroupsApplications of Lagrange Theorem
Subgroups of Prime Index
Overview
The index multiplication formula has a sharp consequence when the index is prime. If a subgroup has prime index in a finite group , then no subgroup can sit strictly between and . This is a structural application of Lagrange's theorem: the index from to cannot split into two nontrivial positive factors. In this lecture, we prove the result and use it to identify when a subgroup is maximal with respect to inclusion.
DEFINITION : Subgroup of Prime Index
Let be a finite group and let be a subgroup of . The subgroup is called a if
for some prime number .
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Author
Dr. Bivash Majumder
Assistant Professor in Mathematics
Prabhat Kumar College, Contai