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".

例題 · 12 mod 5

Number-Theory · Modular-Arithmetic

12 mod 5

12,\ 5

ステップバイステップ

  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).

答えを表示
12 \bmod 5 = 2

今度はあなたの番 問題を選択し、タイプしたり描いたりしてください。 すべてのステップは絵で、答えはあなたが尋ねるまで隠されています。

自分の解答を記録しよう

無料アカウントでは、すべてのレッスンにノートを追加し、完成したことの記録、解いた問題を一つの場所に保存し、このページについて質問できる先生を追加します。数学自体はログインしたかどうかに関係なく誰でも利用できます。

登録 ログイン

ここで使用された記号

どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。

どうやって: 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).

よくある質問

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