Re: [math-fun] Amusing Permutation Representations of Group Extensions
Last line had some numbers wrong but is now fixed below. —Dan ----- Nice pictures in that paper! (I know nothing about group representations, and don't even know the definition of a wreath product.) Funny, just last night I was musing about how for any finite group G the fewest number N = N(G) of points on which it can act faithfully is an interesting invariant. Which is just finding G as a subgroup of the permutation group S_N. For instance, what is this number for the rotation groups of the regular polyhedra: T (order 12), O (order 24), I (order 60) ? Each group acts faithfully on the face centers, edge midpoints, and also the face centers of the dual polyhedron (for O and I). For the octahedron and icosahedron, and for the edges of the tetrahedron, antipodal points get rotated to antipodal points, so we can divide all these numbers by 2 except for the 4 face centers of the tetrahedron. Hence T permutes 4 or 3 = 6/2 points, O permutes 4 = 8/2 or 6 = 12/2 or 3 = 6/2 points, and I permutes 10 = 20/2 or 15 = 30/2 or 6 = 12/2 points. But wait! We can't represent T or O faithfully on 3 points! S_3 is only order 6, while these groups are bigger. Which of these 8 cases are faithful (represent all group elements as distinct permutations)? Can we do better than 4 for T (clearly not), 4 for O, and 6 for I ? -----
For any group G, we are interested in the smallest k for which there exists an endomorphism from G to Sym(k). Call this the "degree" of G. I thought it was classical that the degree of the icosahedral group was 5. I think I've seen an edge-coloring of the dodecahedron that illustrated this: every face shows a different cyclic ordering of the same 5 colors, so every rotation of the solid induces a permutation of the colors. I have been interested, on and off, in the following question: what is the maximal degree Dmax(n) of all groups of order n? Clearly when n in prime, Dmax(n)=n; I'm pretty sure this is true for prime powers in general. I'm also reasonably sure that Dmax(6) = 5. I have the feeling that Dmax(n) can't be _much_ smaller than n, but it's only a feeling. On Fri, Dec 28, 2018 at 1:09 PM Dan Asimov <dasimov@earthlink.net> wrote:
Last line had some numbers wrong but is now fixed below. —Dan
----- Nice pictures in that paper! (I know nothing about group representations, and don't even know the definition of a wreath product.)
Funny, just last night I was musing about how for any finite group G the fewest number N = N(G) of points on which it can act faithfully is an interesting invariant. Which is just finding G as a subgroup of the permutation group S_N.
For instance, what is this number for the rotation groups of the regular polyhedra: T (order 12), O (order 24), I (order 60) ?
Each group acts faithfully on the face centers, edge midpoints, and also the face centers of the dual polyhedron (for O and I).
For the octahedron and icosahedron, and for the edges of the tetrahedron, antipodal points get rotated to antipodal points, so we can divide all these numbers by 2 except for the 4 face centers of the tetrahedron.
Hence T permutes 4 or 3 = 6/2 points, O permutes 4 = 8/2 or 6 = 12/2 or 3 = 6/2 points, and I permutes 10 = 20/2 or 15 = 30/2 or 6 = 12/2 points.
But wait! We can't represent T or O faithfully on 3 points! S_3 is only order 6, while these groups are bigger. Which of these 8 cases are faithful (represent all group elements as distinct permutations)?
Can we do better than 4 for T (clearly not), 4 for O, and 6 for I ? -----
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The following discussion on mathoverflow is helpful https://mathoverflow.net/questions/16858/smallest-permutation-representation... On Fri, Dec 28, 2018 at 13:39 Allan Wechsler <acwacw@gmail.com> wrote:
For any group G, we are interested in the smallest k for which there exists an endomorphism from G to Sym(k). Call this the "degree" of G.
I thought it was classical that the degree of the icosahedral group was 5. I think I've seen an edge-coloring of the dodecahedron that illustrated this: every face shows a different cyclic ordering of the same 5 colors, so every rotation of the solid induces a permutation of the colors.
I have been interested, on and off, in the following question: what is the maximal degree Dmax(n) of all groups of order n? Clearly when n in prime, Dmax(n)=n; I'm pretty sure this is true for prime powers in general. I'm also reasonably sure that Dmax(6) = 5. I have the feeling that Dmax(n) can't be _much_ smaller than n, but it's only a feeling.
On Fri, Dec 28, 2018 at 1:09 PM Dan Asimov <dasimov@earthlink.net> wrote:
Last line had some numbers wrong but is now fixed below. —Dan
----- Nice pictures in that paper! (I know nothing about group representations, and don't even know the definition of a wreath product.)
Funny, just last night I was musing about how for any finite group G the fewest number N = N(G) of points on which it can act faithfully is an interesting invariant. Which is just finding G as a subgroup of the permutation group S_N.
For instance, what is this number for the rotation groups of the regular polyhedra: T (order 12), O (order 24), I (order 60) ?
Each group acts faithfully on the face centers, edge midpoints, and also the face centers of the dual polyhedron (for O and I).
For the octahedron and icosahedron, and for the edges of the tetrahedron, antipodal points get rotated to antipodal points, so we can divide all these numbers by 2 except for the 4 face centers of the tetrahedron.
Hence T permutes 4 or 3 = 6/2 points, O permutes 4 = 8/2 or 6 = 12/2 or 3 = 6/2 points, and I permutes 10 = 20/2 or 15 = 30/2 or 6 = 12/2 points.
But wait! We can't represent T or O faithfully on 3 points! S_3 is only order 6, while these groups are bigger. Which of these 8 cases are faithful (represent all group elements as distinct permutations)?
Can we do better than 4 for T (clearly not), 4 for O, and 6 for I ? -----
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Dan politely informed me that I got some terminology wrong. I wanted a single word for "injective homomorphism" and produced "endomorphism", which apparently isn't required to be 1-1. The word I was looking for is "monomorphism". Sorry for the confusion! On Fri, Dec 28, 2018, 1:45 PM Victor Miller <victorsmiller@gmail.com wrote:
The following discussion on mathoverflow is helpful
https://mathoverflow.net/questions/16858/smallest-permutation-representation...
On Fri, Dec 28, 2018 at 13:39 Allan Wechsler <acwacw@gmail.com> wrote:
For any group G, we are interested in the smallest k for which there exists an endomorphism from G to Sym(k). Call this the "degree" of G.
I thought it was classical that the degree of the icosahedral group was
I think I've seen an edge-coloring of the dodecahedron that illustrated this: every face shows a different cyclic ordering of the same 5 colors, so every rotation of the solid induces a permutation of the colors.
I have been interested, on and off, in the following question: what is the maximal degree Dmax(n) of all groups of order n? Clearly when n in prime, Dmax(n)=n; I'm pretty sure this is true for prime powers in general. I'm also reasonably sure that Dmax(6) = 5. I have the feeling that Dmax(n) can't be _much_ smaller than n, but it's only a feeling.
On Fri, Dec 28, 2018 at 1:09 PM Dan Asimov <dasimov@earthlink.net> wrote:
Last line had some numbers wrong but is now fixed below. —Dan
----- Nice pictures in that paper! (I know nothing about group representations, and don't even know the definition of a wreath product.)
Funny, just last night I was musing about how for any finite group G the fewest number N = N(G) of points on which it can act faithfully is an interesting invariant. Which is just finding G as a subgroup of the permutation group S_N.
For instance, what is this number for the rotation groups of the regular polyhedra: T (order 12), O (order 24), I (order 60) ?
Each group acts faithfully on the face centers, edge midpoints, and also the face centers of the dual polyhedron (for O and I).
For the octahedron and icosahedron, and for the edges of the tetrahedron, antipodal points get rotated to antipodal points, so we can divide all these numbers by 2 except for the 4 face centers of the tetrahedron.
Hence T permutes 4 or 3 = 6/2 points, O permutes 4 = 8/2 or 6 = 12/2 or 3 = 6/2 points, and I permutes 10 = 20/2 or 15 = 30/2 or 6 = 12/2 points.
But wait! We can't represent T or O faithfully on 3 points! S_3 is only order 6, while these groups are bigger. Which of these 8 cases are faithful (represent all group elements as distinct permutations)?
Can we do better than 4 for T (clearly not), 4 for O, and 6 for I ? -----
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participants (3)
-
Allan Wechsler -
Dan Asimov -
Victor Miller