1993 Paper 3 Q5

Year: 1993
Paper: 3
Question Number: 5

Course: UFM Additional Further Pure
Section: Groups

Difficulty: 1730.5 Banger: 1466.6

groups binary operation associativity commutativity complex numbers subgroup isomorphism order of elements identity element inverse matrix quaternion group

Problem

The set $S$ consists of ordered pairs of complex numbers $(z_1,z_2)$ and a binary operation $\circ$ on $S$ is defined by $$ (z_1,z_2)\circ(w_1,w_2)= (z_1w_1-z_2w^*_2, \; z_1w_2+z_2w^*_1). $$ Show that the operation $\circ$ is associative and determine whether it is commutative. Evaluate $(z,0)\circ(w,0)$, $(z,0)\circ(0,w)$, $(0,z)\circ(w,0)$ and $(0,z)\circ(0,w)$. The set $S_1$ is the subset of $S$ consisting of $A$, $B$, $\ldots\,$, $H$, where $A=(1,0)$, $B=(0,1)$, $C=(i,0)$, $D=(0,i)$, $E=(-1,0)$, $F=(0,-1)$, $G=(-i,0)$ and $H=(0,-i)$. Show that $S_1$ is closed under $\circ$ and that it has an identity element. Determine the inverse and order of each element of $S_1$. Show that $S_1$ is a group under $\circ$. \hfil\break [You are not required to compute the multiplication table in full.] Show that $\{A,B,E,F\}$ is a subgroup of $S_1$ and determine whether it is isomorphic to the group generated by the $2\times2$ matrix $\begin{pmatrix}0 & 1\\ -1 & 0 \end{pmatrix}$ under matrix multiplication.

No solution available for this problem.

Rating Information

Difficulty Rating: 1730.5

Difficulty Comparisons: 2

Banger Rating: 1466.6

Banger Comparisons: 4

Show LaTeX source

Problem source

The set $S$ consists of ordered pairs of complex numbers
$(z_1,z_2)$ and a binary operation  $\circ$ on $S$  is defined by
$$
(z_1,z_2)\circ(w_1,w_2)=
(z_1w_1-z_2w^*_2, \; z_1w_2+z_2w^*_1).
$$
Show that the operation $\circ$ is associative and determine whether
it is commutative. Evaluate $(z,0)\circ(w,0)$, $(z,0)\circ(0,w)$,
$(0,z)\circ(w,0)$ and $(0,z)\circ(0,w)$.
The set $S_1$ is the subset of $S$ consisting of $A$, $B$, $\ldots\,$, $H$,
where $A=(1,0)$, $B=(0,1)$, $C=(i,0)$, $D=(0,i)$, $E=(-1,0)$, $F=(0,-1)$,
$G=(-i,0)$ and $H=(0,-i)$. Show that $S_1$ is closed under $\circ$ and 
that it has an identity element. Determine the inverse and order of
each element of $S_1$. Show that $S_1$ is a group under 
$\circ$.
\hfil\break
[You are not required to compute the multiplication table
in full.]
Show that $\{A,B,E,F\}$ is a subgroup of $S_1$ and determine whether
it is isomorphic to the group generated by the $2\times2$ matrix
$\begin{pmatrix}0 & 1\\
-1 & 0
\end{pmatrix}$ under 
matrix multiplication.

Back to List