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November 6, 2017

TIFR 2014 Problem 20 Solution - Symmetries of Cube


TIFR 2014 Problem 20 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Abstract Algebra by David S. Dummit and Richard M. Foote. This book is very useful for the preparation of TIFR Entrance.

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Problem


Let (C) denote the cube ([-1,1]^3\subset \mathbb{R}^3 ). How many rotations are there in (\mathbb{R}^3) which take (C) to itself?


Discussion:


Let us label the six faces of the cube by (F_1,F_2,...,F_6).

Let (G) be the set consisting of all the rotations of (\mathbb{R}^3) which take (C) to itself. This set forms a group under the binary operation composition. In fact, (G) is a subgroup of the group of isometries of (\mathbb{R}^3). The identity is the identity transformation (rotation by 0 degree).

This group (G) acts on the set {(F_1,...F_6)}. The action is the most obvious one. A face (F_i) goes to face (F_j) under any rotation that preserves (C). That is, every rotation is a permutation of the faces (F_1,...,F_6). Said in other words: (\sigma . F_i = F_j) is the action where the face (F_i) goes to (F_j) under the rotation (\sigma).

We recall the Orbit-Stabilizer theorem:

( |G|=|orbit(x)||stab(x)| ). Where (G) acts on the set (X) and (x\in X).

Where (orbit(x)={ g.x | g\in G}) and (stab(x)={g\in G | g.x=x } ) is the stabilizer of (x). Here "." denotes the action.

 

In this context, we have our group (G) as described above. And (X={F_1,...,F_6}).

In order to apply the orbit-stabilizer theorem, we look at (orbit(F_1)) and (stab(F_1)).

We can always rotate the cube in order that the face (F_1) goes to the face (F_2,..,F_6) and also the zero-rotation takes (F_1) to (F_1). Therefore (orbit(F_1)={F_1,...,F_6}).

What is the stabilizer of (F_1)? Well, the only rotations which fix the face (F_1) are the (0,\pi/2,2\pi/2,3\pi/2) rotations in the plane of (F_1) (i.e, having axis perpendicular to (F_1). Hence (|stab(F_1)|=4).

By orbit-stabilizer theorem we now have (|G|=6\times 4=24 ).


Helpdesk

  • What is this topic:Abstract Algebra
  • What are some of the associated concept: Orbit-Stabilizer Theorem
  • Book Suggestions: Abstract Algebra by David S. Dummit and Richard M. Foote.

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