SUMMARY
If G is a finite group with exactly one subgroup H of a given order, then H is normal in G. This conclusion follows from the properties of cosets, where the left cosets and right cosets of H in G coincide. The proof relies on the definition of normal subgroups and the unique existence of H, leading to the conclusion that for any element x in G, the conjugate xHx-1 equals H.
PREREQUISITES
- Understanding of finite group theory
- Familiarity with subgroup definitions and properties
- Knowledge of cosets and their significance in group theory
- Basic concepts of normal subgroups
NEXT STEPS
- Study the definition and properties of normal subgroups in group theory
- Learn about cosets and their role in understanding subgroup structures
- Explore examples of finite groups and their subgroups
- Investigate the implications of the Lagrange's theorem in group theory
USEFUL FOR
Mathematics students, group theory researchers, and anyone interested in the properties of finite groups and subgroup normality.