This page tracks the theorem statements you want to be able to recall cleanly under pressure.

Current Entries

• Subgroup Test: the $x{y}^{-1}$ criterion for subgroup membership.
• Cayley’s Theorem: every group embeds into a symmetric group.
• Lagrange’s Theorem: subgroup order divides group order in the finite case.
• Structure Theorem for Finitely Generated Abelian Groups: classification by free rank and cyclic torsion factors.
• Kernel Criterion for Injectivity: a homomorphism is injective exactly when its kernel is trivial.
• Quotient Group Criterion: coset multiplication is well defined exactly when the subgroup is normal.