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)

Comments

The phrase "double module" in the setting of 2) is obsolete.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 $.