arkajad said:Moreover, this is a frequent notation. See for instance this online course: http://user.math.uzh.ch/halbeisen/4students/gt.html"
Go to "Subgroups"
Newtime said:In fact, I've never seen this notation not used.
Subgroup notation is a way to represent subgroups within a larger group. It typically consists of a symbol or letter that represents the subgroup, followed by the larger group in which it is contained, separated by a vertical bar. For example, the subgroup notation for a subgroup H within a group G would be written as H|G.
Subgroup notation is used in abstract algebra, a branch of mathematics that studies groups, rings, and fields. It is used to represent the different subgroups within a larger group and to perform operations on these subgroups. It allows for a more concise and organized way of discussing and analyzing group structures.
The vertical bar in subgroup notation serves as a separator between the subgroup and the larger group. It indicates that the subgroup is contained within the larger group and helps to distinguish it from other elements or subgroups within the group.
Subgroup notation and cosets are closely related concepts. Cosets are the different equivalence classes of a subgroup within a larger group, and subgroup notation is used to represent these cosets. Each coset corresponds to a different left or right subgroup of the larger group, and subgroup notation helps to keep track of these different subgroups.
Yes, subgroup notation can be used for all types of groups, including finite groups, infinite groups, abelian groups, and non-abelian groups. It is a universal notation that is used in abstract algebra to represent and analyze the subgroups within any type of group structure.