Abstract Algebra 39: A proof of Lagrange's theorem

Abstract Algebra 39: A proof of Lagrange's theorem

Abstract: Lagrange's theorem states that if G is a finite group and H is a subgroup of G, then the size of H divides evenly into the size of G. We prove Lagrange's theorem using properties about the cosets of H in G (that we described in the prior video).

This video accompanies the class "Introduction to Abstract Algebra" at Colorado State University:

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I think you did not demonstrate sufficiently that G = a_1 H U a_2 H U ... a_r H,
since it is conceivable that U { a_i H }, i = 1.. r, may not contain all the elements of the group G.

Specifically your proof breaks down, for example in Z / 12Z for the subgroup <3> = { 3, 6, 9, 0 },
considering the set of distinct left cosets { 0 + <3>, 1 + <3>, 2 + <3> }. The rest of the left cosets a + < 3 > are not distinct .
Clearly 0, 1, 2 are in the union ( 0 + <3> ) U ( 1 + <3> ) U (2 + <3> ), since <3> contains the identity 0.
But what about such numbers as 4, since the left coset 4 + <3> is not in that union.
It is conceivable that 4 is not in that union. Thus { 0 + <3> , 1 + <3> , 2 + <3> } is not equal to Z/12Z, conceivably.

It is equal to Z/12Z, but i feel like you haven't proved it sufficiently.
Great video!