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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".
Çözümlü örnek · 12 mod 5
Adım adım
- 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).
Cevabı göster
Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.
Burada kullanılan sembolleri
Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.
Nasıl: 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).
Sık sorulan sorular
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.
More in Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsRings and fieldsGalois theory: why the quintic has no formula