Proof: Cosets are Disjoint and Equal Size

Explanation for why cosets of a subgroup are either equal or disjoint and why all cosets have the same size.

0:00 Cosets are disjoint
3:15 Cosets have same size

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Music: C418 - Pr Department

[blackpenredpen]

Welcome back!
And wow abstract algebra!!

[thisisanashmonttrain]

Dude where were you for a long time, good to see you.

[StNick]

Lovely relaxing vid. Funnily enough, I've never seen these proofs during my undergrad.

[pauselab]

another important fact is that any element of a group is included inside a coset of one of it's subgroups. if H is a subgroup of G and g an element of G, then g is an element of the coset gH (because H contains the identity). so every element g of G is included in a coset and no element g is in two cosets. with the fact that each coset has the same order as H, it means that the order of G is equal to the index of H in G multiplied by the order of H. |G|=[G:H] * |H|

[manthing]

LoL, i've been watching some of your older videos, and totaly forgot you have a beard now.😅
Nice video.🙂

[alejrandom]

Sounds like the dot product of orthonormal basis vectors. Either 0 if they are distinct or 1 if they are the same... is there any relation?

[erikestrella]

Que cosa tan chida c': recuerdos llegan a mi mente

[taran]

you could expand this even further by claiming there is a bijection between H and aH