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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.
Çözümlü örnek · divisors of 12
Adım adım
- 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.
Cevabı göster
Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.
Burada kullanılan sembolleri
Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.
Nasıl: 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.
Sık sorulan sorular
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
GroupsCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula