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".
हल किया गया उदाहरण · 12 mod 5
चरण दर चरण
- 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).
जवाब दिखाएँ
अब आप कोई प्रश्न चुनें, या अपना प्रश्न टाइप करें या बनाएं। हर चरण, एक चित्र, और उत्तर तब तक छिपा रहता है जब तक आप न पूछें।
एक स्वतंत्र वृत्तान्त प्रत्येक पाठ पर आगे कहता है, जो आपने पूरा किया है, आपके हल किये गये समस्याओं का रिकॉर्ड, और एक शिक्षक जिसके बारे में आप पूछ सकते हैं ।
साइन अप करें लॉगिनयहाँ उपयोग में संकेत
पूरी परिभाषा, चित्र, और इसमें हर अक्षर का अर्थ क्या है?
कैसे: 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).
लोग जो सवाल पूछते हैं
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