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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".

Kugwira ntchito chitsanzo · 12 mod 5

Number-Theory · Modular-Arithmetic

12 mod 5

12,\ 5

Gawo ndi Gawo

  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).

Kusonyeza yankho
12 \bmod 5 = 2

Tsopano inu Sankhani vuto, kapena kulemba kapena kujambula wanu. Pakalipano, yankho limeneli lidzapezeka patsamba lino.

Kusunga ntchito yanu yokha

M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.

Kulembetsa Kulowa

Zithunzi

Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.

Momwe mungagwiritsire ntchito: 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).

Mafunso omwe anthu amafunsa

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.

Zambiri pa Abstract Algebra

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