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".
Разобранный пример · 12 mod 5
Шаг за шагом
- 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).
Показать ответ
Теперь ты Pick a problem, or type or draw your own. Every step, a picture, the answer hidden until you ask.
Бесплатный аккаунт добавляет записи на каждом уроке, записи о том, что вы закончили, ваши проблемы решены в одном месте, и репетитор, которого вы можете спросить об этой странице. Сама математика открыта для всех, подписана или нет.
Зарегистрироваться ВходИспользуемые здесь символы
Нарисуйте любой символ для полного определения, картинку, и что каждая буква в ней означает.
Как: 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).
Вопросы, которые люди задают
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.
Ещё в разделе Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsRings and fieldsGalois theory: why the quintic has no formula