# Double module

A synonym for bimodule.

A pair of subgroups $H, F$ of a group $G$ which are members of the decomposition of $G$ into double cosets, i.e. in the decomposition of $G$ into non-intersecting subsets of the type $H x F$, where $x$ is an element of $G$. A subset $H x F$ is said to be a coset of the group $G$ modulo $( H , F )$ or a double coset of the group $G$ modulo $( H , F )$. Thus, the decomposition of a group of order 24 into double cosets modulo $( H , F )$, where $H$ and $F$ are its Sylow $2$- and $3$- subgroups, consists of a single coset modulo $( H , F )$. Any double coset $H x F$ consists of $| H: H \cap xF x ^ {-} 1 |$ cosets of $G$ by $F$ and, at the same time, of $| F: F \cap x ^ {-} 1 Hx |$ cosets of $G$ by $H$, where $| U: V |$ denotes the index of a subgroup $V$ in a group $U$.

#### References

 [1] P. Hall, "The theory of groups" , Macmillan (1959)

The phrase "double module" in the setting of 2) is obsolete. One uses instead the phrase "double cosetdouble coset" . The double cosets of $G$ modulo $( H , F )$ coincide with the orbits of the direct product $H \times F$ in $G$, acting by $( h , f ) g = h g f ^ { - 1 }$, $h \in H$, $f \in F$, $g \in G$. (See also Orbit). The set of these double cosets is denoted by $H \setminus G / F$.