Contando quadrados coloridos com o Lema que não e de Burnside

Neste quadrados utilizando palitos de picol´e que tenham quadrados que podemos que palitos O processo de contagem dos quadrados coloridos envolve as rota¸c˜oes e reflex˜oes que preservam o quadrado, chamadas simetrias. No primeiro caso o conjunto de simetrias que consideramos consiste apenas das rota¸c˜oes que preservam o quadrado enquanto, no segundo caso, devemos considerar reflex˜oes. Em cada caso, com a composi¸c˜ao de fun¸c˜oes, o conjunto de simetrias que consideramos constitui uma estrutura alg´ebrica denominada grupo. No decorrer do artigo, apresentaremos o conceito de a¸c˜ao de grupo e um resultado relacionado que, entre v´arias denomina¸c˜oes, conhecido como o Lema que n˜ao ´e de Burnside. Abstract In this paper we count how many colored squares may be constructed with popsicle sticks painted, each one, with one of m colors. Two situations are considered, we paint one or both sides of each stick. We will see that in order to construct a square with sticks that have both sides painted, with the same color, we use more paint, however, the number of squares that may be constructed is smaller than using sticks with only one side painted. The method used to count the coloured squares involves rotations and reflections that preserve the square, called symmetries. In the first case the symmetries that we consider are the rotations that preserve the square while, in the second, case we also need to consider reflections. In each case, with the composition of functions, the set of symmetries considered forms an algebraic structure known as group. Throughout the paper we present the notion of group actions and a related result that, among many eponymous, became known also as The Lemma that is not Burnside’s.