We call left loop a magma $(L,\cdot)$ such that
- for all $(a,b)\in L\times L$, exists only one $x\in L$ such that $a\cdot x=b$,
- exists one $e\in L$ such that $e\cdot x=x=x\cdot e$ for all $x\in L$.
This is, a left loop is a left quasigroup with a two-sided identity.
This page of italian wikipedia shows (without citing any reference) what I summarize in the following lines.
Let $G$ be a group (not necessarily commutative), $H$ a subgroup of $G$ (not necessarily normal), $G/H$ the set of the left cosets of $G$ mod $H$. Let $\sigma$ be a function $\sigma\colon G/H\to G$ such that
- $\sigma(H)$ is the identity element in $G$,
- $\sigma$ associates to each left coset one of its own representatives.
Then (Theorem 1), the set $\sigma(G/H)$ has a structure of left loop respect to the operation $\cdot$ defined: $$a\cdot b:=\sigma(abH).$$
I am struggling, time to time, to find any references for that page and to find any book or paper showing this construction and discussing this topic more extensively than I and it.wikipedia did.
Any suggestion of references for this theorem is welcome if you know one, and suggestions of books or papers facing left loop structure in any context are also welcome. Thank you.
Addendum: to make it a bit more interesting and to improve my reference research, I add: does the study of left loops originally arise from (left) cosets?
The following may be of help: http://www.ams.org/journals/tran/1939-046-00/S0002-9947-1939-0000035-5/S0002-9947-1939-0000035-5.pdf
Baer shows (among other things!) that every loop can be represented as a transversal in a permutation group. In this setting (since transversal means different things in different "worlds"), we mean a unique representative from each left (right) coset for a given subgroup. If I'm not mistaken, this is precisely what your reference states (and proves!).
Other references that might be of use are: