91Ó°ÊÓ

Determine which of the following sets are groups under the specified operations: (a) the integers under the operation of subtraction; (b) the set \(\mathbf{R}\) of real numbers under the operation o given by \(a \circ b=a+b+2\) (c) the set of odd integers under the operation of multiplication; (d) the set of \(n \times n\) real matrices whose determinant is either 1 or \(-1\), under matrix multiplication.

Calculate the multiplication table for the following eight \(2 \times 2\) complex matrices, and deduce that they form a non-abelian group: $$ \begin{aligned} &I=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right), A=\left(\begin{array}{rr} i & 0 \\ 0 & -i \end{array}\right), B=\left(\begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array}\right), C=\left(\begin{array}{rr} -i & 0 \\ 0 & i \end{array}\right) \\ &D=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), E=\left(\begin{array}{rr} 0 & i \\ -i & 0 \end{array}\right), F=\left(\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right), G=\left(\begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right) \end{aligned} $$

Find the multiplication table for the eight symmetries of a square.

Find the symmetry groups of (a) a non-square rectangle, (b) a parallelogram with unequal sides which is not a rectangle, (c) a non-square rhombus.

Write down the multiplication tables for the groups \(C_{2} \times C_{3}\) and \(C_{3} \times C_{3}\).

Show that \(G \times H\) is abelian if and only if \(G\) and \(H\) are each abelian.