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The definition I have for left cosets is as follows:

Let $G$ be a group and let $H$ be a subgroup of $G$. A left coset of $H$ in $G$ is a set of the form $gH=\{gh:h\in H\}$ for some $g\in G$.

Right cosets are defined similarly.

Is there any purpose in relaxing the requirement that $H$ be a subgroup so that $H$ only needs to be a subset of $G$? If so, what properties of cosets will still hold when this requirement is relaxed?

The only result I have found so far that still holds is $1H=H=H1$. This doesn't seem like a very useful result on its own.

It also seems that the following results won't hold when the requirement is relaxed:

  1. The cardinality of the set of left cosets is equal to the that of the set of right cosets.
  2. The set of left cosets forms a partition of the group $G$.
  3. Lagrange's Theorem (the proof I have seen uses the the fact that the left cosets form a partition of $G$).
Arctic Char
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Anson
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    Since different cosets won't be disjoint, I'd say that pretty much everything that's true about cosets of a subgroup can be expected to fail. For example, everything that's based on the idea of choosing a representative element of a coset will fall apart. – hmakholm left over Monica Jun 17 '15 at 00:20

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The cosets are defined by the equivalence relation: $g\sim g'\Leftrightarrow g^{-1}g'\in H$. If you consider only subsets of the group $G$, you will not get an equivalence relation, as you can check. As you surmised, you then lose your partition of $G$. So, you really need to use subgroups. Incidently, if you require $H$ to be normal in $G$ the cosets will actually be a group.

Matematleta
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  • That makes sense. Can I ask if there are any textbook(s) you know of that use the equivalence relation definition? I'm using Abstract Algebra by Dummit and Foote, and occasionally Algebra by Sigler. – Anson Jun 17 '15 at 01:11
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    @Anson Hungerford uses the above equivalence relation to define cosets in his algebra textbook. – Taylor Jun 17 '15 at 01:21
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    Herstein's Modern Algebra might be another good choice – Matematleta Jun 17 '15 at 01:30
  • Thanks to both of you. I've already heard good things about both books! – Anson Jun 17 '15 at 01:33
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    You might want to read this: http://math.stackexchange.com/questions/1252108/intuition-behind-quotient-groups/1253636#1253636 – David Wheeler Jun 17 '15 at 01:54