Applying First Isomorphism to φ

Abstract Algebra · Axiom Academy

EXAMPLE First Isomorphism Theorem Applying the theorem to to construct and verify the quotient isomorphism Matching Exercise: Cosets to Image Elements Match each coset in with its corresponding element in . Click pairs to match them. Excellent work! You've successfully applied the First Isomorphism Theorem. Here's what we learned: Finding the Kernel: ( ) = \ x _ 12 : (x) = 0 _4\ = \ 0, 4, 8\ , which forms a subgroup of _ 12 Constructing the Quotient: has exactly 4 cosets, one for each equivalence class modulo the kernel The Isomorphism: The induced map : _ 12 / ( ) ( ) = _4 is both one-to-one and onto, preserving the group operation First Isomorphism Theorem: For any homomorphism , we have G/ ( ) ( ) . In our case, _ 12 /\ 0, 4, 8\ _4 Coset Representatives: Each coset can be represented by any of its elements: 0, 1, 2, 3 serve as canonical representatives This theorem is fundamental in abstract algebra! It tells us that every homomorphism "factors through" a quotient by its kernel, revealing deep structural relationships between groups. You'll see this pattern repeatedly in ring theory, module theory, and beyond!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.