Mathematics
Homomorphic vs Homeomorphic vs Isomorphic
A focused guide to the difference between homomorphic, homeomorphic, and isomorphic, with examples from algebra and topology.
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7 posts tagged "Algebra"
Mathematics
What Is a Morphism? A Clear Guide for Students
A professional introduction to morphisms, homomorphisms, homeomorphisms, isomorphisms, and the general idea of structure-preserving maps across mathematics.
Mathematics
Lagrange's Theorem: Why Subgroup Orders Divide Group Orders
We prove Lagrange's theorem — the fundamental result that the order of a subgroup divides the order of the group — and explore its many consequences, from Fermat's little theorem to the structure theory of finite groups.
Mathematics
The Classification of Finite Simple Groups: The Enormous Theorem
We survey the Classification of Finite Simple Groups — the monumental theorem that catalogs every possible finite simple group — covering its structure, history, and significance as perhaps the longest proof in the history of mathematics.
Mathematics
Noether's Theorem: How Symmetry Creates Conservation Laws
We explain and prove Noether's theorem — the profound result connecting continuous symmetries of a physical system to conservation laws — and explore how it unifies energy, momentum, and angular momentum under a single mathematical principle.
Mathematics
Emmy Noether: The Mother of Modern Algebra
The remarkable story of Emmy Noether, who overcame institutional sexism to transform abstract algebra and whose theorem linking symmetry to conservation laws remains one of the deepest results in mathematical physics.
Mathematics
Alexander Grothendieck: The Visionary Who Rebuilt Mathematics
The story of Alexander Grothendieck — a stateless refugee who became the most revolutionary mathematician of the twentieth century, rebuilt algebraic geometry from the ground up, and then walked away from it all.