User:Comfr/coset

Coset example

Group G = {0,1,2,3,4,5,6,7}

Subgroup H of G = {0,3}

(Z/8Z, +), the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to (Z/2Z, +).

Each element of G has both a left and a right coset, but some elements can be in the same coset, so the number of cosets may be less than the number of elements in G.

Elements g and x belong to the same left coset of H if and only if g-1x belongs to H.

Comments from a talk page
This discussion really belongs on the article's talk page, but I'll give a brief response here. You are correct in pointing out that the article's introduction leaves much to be desired. I could see what you were trying to express, but it came out more confusing than not.

Each element is in a unique (there is only one) coset, but the cosets are not distinct (different from one another). The rule for telling when two cosets are equal is given in the third paragraph, but it would take some familiarity with the subject to glean the consequences of that rule.

Perhaps a better way to see what is going on is to say that the distinct cosets form a partition of $G$ (every element of $G$ is in exactly one piece [coset] of the partition). The illustration shows this partition by listing all of the distinct cosets of $G$. The illustration can also be interpreted as the underlying set of the quotient group $G/H$ (its not a group until you specify how the cosets are to be added).

I've been thinking of how to rewrite the lead of the article and have been toying with the idea of introducing the partition idea early on. Does this make sense to you? --Bill Cherowitzo (talk) 00:06, 17 May 2020 (UTC)