91Ó°ÊÓ

Write out Cayley tables for groups formed by the symmetries of a rectangle and for \(\left(\mathbb{Z}_{4},+\right)\). How many elements are in each group? Are the groups the same? Why or why not?

Describe the symmetries of a rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a rectangle and the symmetries of a rhombus. Are the symmetries of a rectangle and those of a rhombus the same?

Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by \(D_{4}\).

Give a multiplication table for the group \(U(12)\).

Prove that the set of matrices of the form $$ \left(\begin{array}{lll} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right) $$ is a group under matrix multiplication. This group, known as the Heisenberg group, is important in quantum physics. Matrix multiplication in the Heisenberg group is defined by $$ \left(\begin{array}{ccc} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{ccc} 1 & x^{\prime} & y^{\prime} \\ 0 & 1 & z^{\prime} \\ 0 & 0 & 1 \end{array}\right)=\left(\begin{array}{ccc} 1 & x+x^{\prime} & y+y^{\prime}+x z^{\prime} \\ 0 & 1 & z+z^{\prime} \\ 0 & 0 & 1 \end{array}\right) $$

Given the groups \(\mathbb{R}^{*}\) and \(\mathbb{Z}\), let \(G=\mathbb{R}^{*} \times \mathbb{Z}\). Define a binary operation o on \(G\) by \((a, m) \circ(b, n)=(a b, m+n)\). Show that \(G\) is a group under this operation.

Prove or disprove that every group containing six elements is abelian.

Give a specific example of some group \(G\) and elements \(g, h \in G\) where \((g h)^{n} \neq g^{n} h^{n}\).

Give an example of three different groups with eight elements. Why are the groups different?

Show that there are \(n !\) permutations of a set containing \(n\) items.