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".

Umzekelo osebenzelayo · 12 mod 5

Number-Theory · Modular-Arithmetic

12 mod 5

12,\ 5

Inyathelo ngenyathelo

  1. 12 = 2 \times 5 + 2

    Divide 12 by 5: the quotient is 2 and what is left over is the remainder.

  2. 12 \bmod 5 = 2

    The remainder is the answer (always between 0 and m − 1).

Bonisa impendulo
12 \bmod 5 = 2

ngoku Khetha ingxaki, okanye ubhale okanye uzobe yakho. Inyathelo ngalinye, umfanekiso, impendulo ifihlakele de ubuze.

Ukugcina umsebenzi wakho

I akhawunti ekhululekileyo idibanisa amaphetshana kwi ncwadi nganye, irekhodi lento ogqibe ngayo, iingxaki zakho ezisoliweyo kwindawo enye, nomfundi onokuthi ubuze malunga nale phepha. IiMathematiki ngokwazo zivuliwe kubo bonke, bangeniswe okanye hayi.

Bhalisa Igama elithile

Iimpawu ezisetyenziswa apha

Nqakraza nasiphi na isibonakaliso sokuqonda okupheleleyo, umfanekiso, nokuba iileta zonke zithetha ntoni.

Indlela yokusebenza: Homomorphisms, normal subgroups and quotient groups

  1. Divide 12 by 5: the quotient is 2 and what is left over is the remainder.
  2. The remainder is the answer (always between 0 and m − 1).

Imibuzo abantu bebuza

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.

IiNkqubo Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsRings and fieldsGalois theory: why the quintic has no formula