Proper condition on the dihedral group
Is there a theream which is a condition on $ninmathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups?
After spending some time figuring a condition I tried to find some similar thread but didn't find any.
group-theory dihedral-groups
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
asked Dec 1 '18 at 13:07
vesii
685
add a comment |
Is there a theream which is a condition on $ninmathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups?
After spending some time figuring a condition I tried to find some similar thread but didn't find any.
group-theory dihedral-groups
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
asked Dec 1 '18 at 13:07
vesii
685
1
When $n$ is composite.
– user10354138
Dec 1 '18 at 13:22
add a comment |
Is there a theream which is a condition on $ninmathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups?
After spending some time figuring a condition I tried to find some similar thread but didn't find any.
group-theory dihedral-groups
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
asked Dec 1 '18 at 13:07
vesii
685
Is there a theream which is a condition on $ninmathbb N$ that says when the dihedral group, $D_{n}$, has non-cyclic subgroups?
After spending some time figuring a condition I tried to find some similar thread but didn't find any.
group-theory dihedral-groups
group-theory dihedral-groups
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
asked Dec 1 '18 at 13:07
vesii
685
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
asked Dec 1 '18 at 13:07
vesii
685
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
edited Dec 1 '18 at 13:38
Jean-Claude Arbaut
14.7k63464
14.7k63464
asked Dec 1 '18 at 13:07
vesii
685
asked Dec 1 '18 at 13:07
vesii
685
asked Dec 1 '18 at 13:07
vesii
685
685
1
When $n$ is composite.
– user10354138
Dec 1 '18 at 13:22
add a comment |
1
When $n$ is composite.
– user10354138
Dec 1 '18 at 13:22
1
1
When $n$ is composite.
– user10354138
Dec 1 '18 at 13:22
When $n$ is composite.
– user10354138
Dec 1 '18 at 13:22
add a comment |
2 Answers
2
active
oldest
votes
Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, forall n ge3$.
By your question, $D_n le D_n$. So true for $forall nge3$.
But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$.
Hence when $n$ is composite we get a required subgroup.
answered Dec 1 '18 at 13:37
Offlaw
2649
add a comment |
Any cyclic group has unique subgroup of any order which is divisior of its order
Now in $D_n$ How many subgroup of order 2?
They are at least n .
SO for n>1
answered Dec 1 '18 at 14:04
Shubham
1,5851519
add a comment |
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2 Answers
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2 Answers
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Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, forall n ge3$.
By your question, $D_n le D_n$. So true for $forall nge3$.
But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$.
Hence when $n$ is composite we get a required subgroup.
answered Dec 1 '18 at 13:37
Offlaw
2649
add a comment |
Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, forall n ge3$.
By your question, $D_n le D_n$. So true for $forall nge3$.
But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$.
Hence when $n$ is composite we get a required subgroup.
answered Dec 1 '18 at 13:37
Offlaw
2649
add a comment |
Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, forall n ge3$.
By your question, $D_n le D_n$. So true for $forall nge3$.
But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$.
Hence when $n$ is composite we get a required subgroup.
answered Dec 1 '18 at 13:37
Offlaw
2649
Dihedral group $D_n = <r, s | r^n = s^2 = 1, rs = sr^{-1}>, forall n ge3$.
By your question, $D_n le D_n$. So true for $forall nge3$.
But if we want a proper non-cyclic subgroup, then we have to consider some $<r^a,s>$.
Hence when $n$ is composite we get a required subgroup.
answered Dec 1 '18 at 13:37
Offlaw
2649
answered Dec 1 '18 at 13:37
Offlaw
2649
answered Dec 1 '18 at 13:37
Offlaw
2649
answered Dec 1 '18 at 13:37
Offlaw
2649
2649
add a comment |
add a comment |
Any cyclic group has unique subgroup of any order which is divisior of its order
Now in $D_n$ How many subgroup of order 2?
They are at least n .
SO for n>1
answered Dec 1 '18 at 14:04
Shubham
1,5851519
add a comment |
Any cyclic group has unique subgroup of any order which is divisior of its order
Now in $D_n$ How many subgroup of order 2?
They are at least n .
SO for n>1
answered Dec 1 '18 at 14:04
Shubham
1,5851519
add a comment |
Any cyclic group has unique subgroup of any order which is divisior of its order
Now in $D_n$ How many subgroup of order 2?
They are at least n .
SO for n>1
answered Dec 1 '18 at 14:04
Shubham
1,5851519
Any cyclic group has unique subgroup of any order which is divisior of its order
Now in $D_n$ How many subgroup of order 2?
They are at least n .
SO for n>1
answered Dec 1 '18 at 14:04
Shubham
1,5851519
answered Dec 1 '18 at 14:04
Shubham
1,5851519
answered Dec 1 '18 at 14:04
Shubham
1,5851519
answered Dec 1 '18 at 14:04
Shubham
1,5851519
1,5851519
add a comment |
add a comment |
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1
When $n$ is composite.
– user10354138
Dec 1 '18 at 13:22