# Double module

A synonym for bimodule.

A pair of subgroups of a group which are members of the decomposition of into double cosets, i.e. in the decomposition of into non-intersecting subsets of the type , where is an element of . A subset is said to be a coset of the group modulo or a double coset of the group modulo . Thus, the decomposition of a group of order 24 into double cosets modulo , where and are its Sylow - and -subgroups, consists of a single coset modulo . Any double coset consists of cosets of by and, at the same time, of cosets of by , where denotes the index of a subgroup in a group .

#### 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The phrase "double module" in the setting of 2) is obsolete.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 modulo coincide with the orbits of the direct product in , acting by , , , . (See also Orbit). The set of these double cosets is denoted by .

**How to Cite This Entry:**

Double module. V.D. Mazurov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Double_module&oldid=13464