maths.free › Abstract Algebra › Homomorphisms, normal subgroups and quotient groups
Homomorphisms, normal subgroups and quotient groups
Structure-preserving maps, kernels, and the first isomorphism theorem.
A homomorphism respects the operation; its kernel is a normal subgroup, and G/ker ≅ image. Picture it: reducing an integer mod 5 collapses the number line onto a 5-cycle. Think it: the isomorphism theorems are the group-theoretic form of "quotient by what you ignore".
Voorbeeld van werk · 12 mod 5
Stap voor stap
- 12 = 2 \times 5 + 2
Divide 12 by 5: the quotient is 2 and what is left over is the remainder.
- 12 \bmod 5 = 2
The remainder is the answer (always between 0 and m − 1).
Onthul het antwoord
Nu jij. Kies een probleem, type of teken je eigen. Elke stap, een foto, het antwoord verborgen totdat je het vraagt.
Hier gebruikte symbolen
Tik op elk symbool voor de volledige definitie, een afbeelding en wat elke letter erin betekent.
Hoe: Homomorphisms, normal subgroups and quotient groups
- Divide 12 by 5: the quotient is 2 and what is left over is the remainder.
- The remainder is the answer (always between 0 and m − 1).
Vragen die mensen stellen
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
Meer in Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsRings and fieldsGalois theory: why the quintic has no formula