Difference between revisions of "Subgroup"

Revision as of 10:46, 2 April 2009

Sous-groupe (Fr); Untergruppe (Ge); Subgrupo (Sp); Sottogruppo (It); 部分群 (Ja).

Let G be a group and H a non-empty subset of G. Then H is called a subgroup of G if the elements of H obey the group postulates, i.e. if

The subgroup H is called a proper subgroup of G if there are elements of G not contained in H.

A subgroup H of G is called a maximal subgroup of G if there is no proper subgroup M of G such that H is a proper subgroup of M.

@@ Line 2: / Line 2: @@
-Let G be a group and H a non-empty subset of G. Then H is called a '''subgroup''' of H if the elements of H obey the group postulates.
+Let ''G'' be a [[group]] and ''H'' a non-empty subset of ''G''. Then ''H'' is called a '''subgroup''' of ''G'' if the elements of ''H'' obey the group postulates, i.e. if
+# the identity element ''1<sub>G</sub>'' of ''G'' is contained in ''H'';
+# ''H'' is closed under the group operation (inherited from ''G'');
+# ''H'' is closed under taking inverses.
-The subgroup H is called a ''proper subgroup'' of G if there are elements of G not contained in H.
+The subgroup ''H'' is called a '''proper subgroup''' of ''G'' if there are elements of ''G'' not contained in ''H''.
-A subgroup H of G is called a ''maximal subgroup'' of G if there is no proper subgroup M of G such that H is a proper subgroup of M.
+A subgroup ''H'' of ''G'' is called a '''maximal subgroup''' of ''G'' if there is no proper subgroup ''M'' of ''G'' such that ''H'' is a proper subgroup of ''M''.
 ==See also==