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Two kinds of of subgroups of a dihedral group

How to find non-cyclic subgroups of a group?Frobenius dihedral groupsFinding Sylow p-subgroupsFinding conjugacy classes and normal subgroups of $D_8$, the dihedral group of order $16$Is there any easier way to find out every proper subgroups of Dihedral group 4?Subgroups of generalized dihedral groupsAre all abelian subgroups of a dihedral group cyclic?Describing all Sylow 2-subgroups of the dihedral group $D_n$Finding Sylow 2-subgroups of the dihedral group $D_n$What dihedral subgroups occur in the affine general linear group $AGL(2,3)$

0

$begingroup$

How do we show that there are two kinds of subgroups of a dihedral group or where can I find a proof of that?

abstract-algebra reference-request

share|cite|improve this question

edited Mar 15 at 5:06

user398843

asked Mar 15 at 4:31

user398843user398843

689316

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Did you just edit this and replace the text with an image? I could've sworn I saw text when I clicked on this in the page listing all the questions, but now it's gone : S
  $endgroup$
  – M. Vinay
  Mar 15 at 4:43
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Anyhow, to answer your question… Any subgroup either contains a reflection (an element of the form $a^r x$) or not. If it does not, then it is the first kind (that is, it is a subgroup of $langle a rangle$). So consider the case where it contains some $a^r x$ and show that it is of the form $langle a^d, a^r x rangle$.
  $endgroup$
  – M. Vinay
  Mar 15 at 4:47
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Hint for that part: Let $H$ be a subgroup containing a reflection $a^r x$ and let $d$ be the least non-negative integer such that $a^d in H$. What if $d = 0$? What if $d ne 0$?
  $endgroup$
  – M. Vinay
  Mar 15 at 4:50
  
  

  
  
  
  

add a comment | 

0

$begingroup$

How do we show that there are two kinds of subgroups of a dihedral group or where can I find a proof of that?

abstract-algebra reference-request

share|cite|improve this question

edited Mar 15 at 5:06

user398843

asked Mar 15 at 4:31

user398843user398843

689316

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Did you just edit this and replace the text with an image? I could've sworn I saw text when I clicked on this in the page listing all the questions, but now it's gone : S
  $endgroup$
  – M. Vinay
  Mar 15 at 4:43
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Anyhow, to answer your question… Any subgroup either contains a reflection (an element of the form $a^r x$) or not. If it does not, then it is the first kind (that is, it is a subgroup of $langle a rangle$). So consider the case where it contains some $a^r x$ and show that it is of the form $langle a^d, a^r x rangle$.
  $endgroup$
  – M. Vinay
  Mar 15 at 4:47
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Hint for that part: Let $H$ be a subgroup containing a reflection $a^r x$ and let $d$ be the least non-negative integer such that $a^d in H$. What if $d = 0$? What if $d ne 0$?
  $endgroup$
  – M. Vinay
  Mar 15 at 4:50
  
  

  
  
  
  

add a comment | 

$begingroup$

How do we show that there are two kinds of subgroups of a dihedral group or where can I find a proof of that?

abstract-algebra reference-request

share|cite|improve this question

edited Mar 15 at 5:06

user398843

asked Mar 15 at 4:31

user398843user398843

689316

$endgroup$

share|cite|improve this question

edited Mar 15 at 5:06

user398843

asked Mar 15 at 4:31

user398843user398843

689316

share|cite|improve this question

edited Mar 15 at 5:06

user398843

asked Mar 15 at 4:31

user398843user398843

689316

asked Mar 15 at 4:31

user398843user398843

689316

asked Mar 15 at 4:31

user398843user398843

689316