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A synonym for [[Bimodule|bimodule]].
 
A synonym for [[Bimodule|bimodule]].
  
A pair of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339001.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339002.png" /> which are members of the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339003.png" /> into double cosets, i.e. in the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339004.png" /> into non-intersecting subsets of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339006.png" /> is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339007.png" />. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339008.png" /> is said to be a coset of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d0339009.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390010.png" /> or a double coset of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390011.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390012.png" />. Thus, the decomposition of a group of order 24 into double cosets modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390015.png" /> are its Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390016.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390017.png" />-subgroups, consists of a single coset modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390018.png" />. Any double coset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390019.png" /> consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390020.png" /> cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390021.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390022.png" /> and, at the same time, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390023.png" /> cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390024.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390026.png" /> denotes the index of a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390027.png" /> in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390028.png" />.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Hall,  "The theory of groups" , Macmillan  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Hall,  "The theory of groups" , Macmillan  (1959)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The phrase  "double module"  in the setting of 2) is obsolete. One uses instead the phrase  "double cosetdouble coset" . The double cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390029.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390030.png" /> coincide with the orbits of the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390032.png" />, acting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390036.png" />. (See also [[Orbit|Orbit]]). The set of these double cosets is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033900/d03390037.png" />.
+
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|Orbit]]). The set of these double cosets is denoted by $  H \setminus  G / F $.

Latest revision as of 19:36, 5 June 2020


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 $.

How to Cite This Entry:
Double module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double_module&oldid=46771
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article