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Subgroups, cosets and Lagrange's theorem
Why the order of a subgroup divides the order of the group.
Cosets of a subgroup H tile the group in equal-sized pieces, so |H| divides |G|. Picture it: the integers mod 12 as a clock; the subgroup {0, 4, 8} and its three cosets colour the clock in three colours. Think it: Lagrange is the reason element orders divide group orders, hence the reason RSA works.
उदाहरण · divisors of 12
चरणद्वारा चरण
- 12 = 2^{2} \times 3
Prime-factorise first.
- d(n) = (2+1) \cdot (1+1) = 6
Each divisor picks an exponent from 0 up to the prime's exponent, so multiply (exponent + 1) for each prime.
- 1, 2, 3, 4, 6, 12
List them in increasing order.
जवाफ प्रकट गर्नुहोस्
अब तपाईँ समस्या रोज्नुहोस्, वा टाइप गर्नुहोस् वा तपाईँको आफ्नै रेखाचित्र गर्नुहोस् । प्रत्येक चरण, एउटा तस्वीर, तपाईँले सोध्नु अघि उत्तर लुकेको छ ।
एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.
साइन अप गर्नुहोस् लगइनयहाँ प्रयोग गरिएको प्रतीक
पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।
कसरी: Subgroups, cosets and Lagrange's theorem
- Prime-factorise first.
- Each divisor picks an exponent from 0 up to the prime's exponent, so multiply (exponent + 1) for each prime.
- List them in increasing order.
प्रश्नहरू मानिसहरूले सोध्छन्
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.
यसमा थप Abstract Algebra
GroupsCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula