Problem #WSP-000301

Problems Geometry Symmetry
 10–16  4.0

Problem

The set of symmetries of an object (e.g. a square) is called a group. We can formally define a group \(G\) as follows.

A is a non-empty set \(G\) with a binary operation \(*:G\times G\to G\) satisfying the following axioms (you can think of them as rules that \(G\) and \(*\) have to satisfy). A binary operation takes two elements of \(G\) and gives another element of \(G\).

  1. Associativity: For all \(g\), \(h\) and \(k\) in \(G\), \((g*h)*k=g*(h*k)\).

  2. Identity: There is an element \(e\) of \(G\) such that \(e*g=g=g*e\) for all \(g\) in \(G\).

  3. Inverses: For every \(g\) in \(G\), there exists a \(g^{-1}\) in \(G\) such that \(g*g^{-1}=e\).

  4. Closure: For all \(g\) and \(h\) in \(G\), \(g*h\) is also in \(G\).

Prove that the symmetries of the ‘reduce-reuse-recycle’ symbol form a group.

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