There is no theorem needed. If you look at the representation ##D_m=\langle r,s\,|\,r^m=s^2=srsr=1 \rangle ## then can you say whether all words over the alphabet ##\{\,r,s\,\}## can be written as ## r^k \cdot 1## or ##r^k\cdot s\,?## Or what do you mean by ##\langle r\rangle \langle s\rangle\,?##AllRelative said:Homework Statement
This is only a step in a proof I am trying to make.
Let Dm be the dihedral group.
r is the rotation of 2π/m around the origin and s is a reflexion about a line passing trough a vertex and the origin.
Let<s> and <r> be two subgroups of Dm.
Is there a theorem that states that Dm = <r><s>
Homework Equations
The Attempt at a Solution
Thanks for the help
fresh_42 said:There is no theorem needed. If you look at the representation ##D_m=\langle r,s\,|\,r^m=s^2=srsr=1 \rangle ## then can you say whether all words over the alphabet ##\{\,r,s\,\}## can be written as ## r^k \cdot 1## or ##r^k\cdot s\,?## Or what do you mean by ##\langle r\rangle \langle s\rangle\,?##
As groups, and if you consider ##\langle r\rangle \langle s\rangle = \langle r\rangle \times \langle s\rangle## as a direct product, then this is not true. One of those subgroups is not normal.
Subgroups of the dihedral group are smaller groups that are contained within the larger dihedral group. They consist of elements from the dihedral group that satisfy the group axioms and form a closed set under the group operation.
The number of subgroups in a dihedral group depends on the order of the group. For a dihedral group of order n, there are n subgroups, including the identity subgroup and the full dihedral group itself.
Subgroups of the dihedral group inherit some properties from the larger group, such as the order of the subgroup being a factor of the order of the dihedral group. They also have their own unique properties, such as being cyclic or non-cyclic.
Subgroups of the dihedral group are closely related to symmetry operations as they are a way of categorizing and organizing the various symmetries that can be performed on a regular polygon. Each subgroup corresponds to a specific set of symmetry operations.
Yes, subgroups of the dihedral group can be useful in simplifying calculations, particularly when dealing with large or complex dihedral groups. By breaking down the larger group into smaller, more manageable subgroups, it becomes easier to analyze and understand the group's structure and properties.