Symmetries of a Hexagon
I am having trouble with the following problem.
Let $D_{12} = <a,b:a^6=b^2=e, ba=a^5b>$, the dihedral group of order 12, describe the symmetries of a regular hexagon. $a$ denotes a rotation about the center by 60 degrees, and $b$ denotes a reflection across a diagonal connecting a pair of opposite vertices.
I need to find the centralizers $C_{D_{12}}(a)$, $C_{D_{12}}(b)$, as well as the center $Z(D_{12})$.
Clearly, each element of $D_{12}$ can be written in the form $a^ib^j$, with $0leq ileq5$, and $0leq jleq1$. To find the centralizers, I think I need to use the Orbit-Stabilizer Theorem, but I'm not sure how to find the center.
abstract-algebra
asked Nov 24 at 19:38
jclay
121
add a comment |
I am having trouble with the following problem.
Let $D_{12} = <a,b:a^6=b^2=e, ba=a^5b>$, the dihedral group of order 12, describe the symmetries of a regular hexagon. $a$ denotes a rotation about the center by 60 degrees, and $b$ denotes a reflection across a diagonal connecting a pair of opposite vertices.
I need to find the centralizers $C_{D_{12}}(a)$, $C_{D_{12}}(b)$, as well as the center $Z(D_{12})$.
Clearly, each element of $D_{12}$ can be written in the form $a^ib^j$, with $0leq ileq5$, and $0leq jleq1$. To find the centralizers, I think I need to use the Orbit-Stabilizer Theorem, but I'm not sure how to find the center.
abstract-algebra
asked Nov 24 at 19:38
jclay
121
add a comment |
I am having trouble with the following problem.
Let $D_{12} = <a,b:a^6=b^2=e, ba=a^5b>$, the dihedral group of order 12, describe the symmetries of a regular hexagon. $a$ denotes a rotation about the center by 60 degrees, and $b$ denotes a reflection across a diagonal connecting a pair of opposite vertices.
I need to find the centralizers $C_{D_{12}}(a)$, $C_{D_{12}}(b)$, as well as the center $Z(D_{12})$.
Clearly, each element of $D_{12}$ can be written in the form $a^ib^j$, with $0leq ileq5$, and $0leq jleq1$. To find the centralizers, I think I need to use the Orbit-Stabilizer Theorem, but I'm not sure how to find the center.
abstract-algebra
asked Nov 24 at 19:38
jclay
121
I am having trouble with the following problem.
Let $D_{12} = <a,b:a^6=b^2=e, ba=a^5b>$, the dihedral group of order 12, describe the symmetries of a regular hexagon. $a$ denotes a rotation about the center by 60 degrees, and $b$ denotes a reflection across a diagonal connecting a pair of opposite vertices.
I need to find the centralizers $C_{D_{12}}(a)$, $C_{D_{12}}(b)$, as well as the center $Z(D_{12})$.
Clearly, each element of $D_{12}$ can be written in the form $a^ib^j$, with $0leq ileq5$, and $0leq jleq1$. To find the centralizers, I think I need to use the Orbit-Stabilizer Theorem, but I'm not sure how to find the center.
abstract-algebra
abstract-algebra
asked Nov 24 at 19:38
jclay
121
asked Nov 24 at 19:38
jclay
121
asked Nov 24 at 19:38
jclay
121
asked Nov 24 at 19:38
jclay
121
asked Nov 24 at 19:38
jclay
121
121
add a comment |
add a comment |
1 Answer
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Centralizer of rotation is all rotations, centralizer of reflection is itself and the identity element. Centralizer of $G$ is identity and rotation by $pi$.
answered Nov 24 at 19:49
mathnoob
1,784422
add a comment |
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1 Answer
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1 Answer
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active
oldest
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Centralizer of rotation is all rotations, centralizer of reflection is itself and the identity element. Centralizer of $G$ is identity and rotation by $pi$.
answered Nov 24 at 19:49
mathnoob
1,784422
add a comment |
Centralizer of rotation is all rotations, centralizer of reflection is itself and the identity element. Centralizer of $G$ is identity and rotation by $pi$.
answered Nov 24 at 19:49
mathnoob
1,784422
add a comment |
Centralizer of rotation is all rotations, centralizer of reflection is itself and the identity element. Centralizer of $G$ is identity and rotation by $pi$.
answered Nov 24 at 19:49
mathnoob
1,784422
Centralizer of rotation is all rotations, centralizer of reflection is itself and the identity element. Centralizer of $G$ is identity and rotation by $pi$.
answered Nov 24 at 19:49
mathnoob
1,784422
answered Nov 24 at 19:49
mathnoob
1,784422
answered Nov 24 at 19:49
mathnoob
1,784422
answered Nov 24 at 19:49
mathnoob
1,784422
1,784422
add a comment |
add a comment |
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