maths.free › Abstract Algebra › Subgroups, cosets and Lagrange's theorem
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.
Delovni primer · divisors of 12
Korak po koraku
- 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.
Odkrij odgovor
Zdaj pa ti. Izberite problem, ali tipkajte ali narišite svojo. Vsak korak, sliko, odgovor skriti, dokler ne vprašate.
Brezplačen račun dodaja opombe o vsaki lekciji, zapis o tem, kar ste končali, vaše rešene težave na enem mestu, in tutor lahko vprašate o tej strani. Sama matematika je odprta za vse, prijavljena ali ne.
Registriraj se PrijavaSimboli, ki se uporabljajo tukaj
Tapnite vsak simbol za popolno opredelitev, sliko in kaj pomeni vsaka črka v njej.
Kako: 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.
Vprašanja, ki jih ljudje sprašujejo
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.
Več v Abstract Algebra
GroupsCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula