a) Let G? be a group with five elements. What are the possible orders of
subgroups of G?? How many elements of G? generate G??
b) Let G? be a group with four elements. What are the possible orders of
subgroups of G??
Let S? = {1, $i$, -1, -$i$}, and
$\begin{aligned} S_2 = \{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \} \end{aligned}$
S? equipped with complex multiplication is a group. S? equipped with ma-
trix multiplication is a group. (You showed this in the last assignment)
c) Show that the two groups are isomorphic.
d) Is S? isomorphic to Z? and/or Z? and/or Z? × Z?? (here addition is the
composition operation)
e) In S? there is one subgroup, H, of order 2. Find its left cosets.
f) Are the left cosets identical to the right cosets? Make at least one argu-
ment that is not direct computation (you may supplement by direct computa-
tion)
g) Is H a normal subgroup? If so, write the Cayley table for the quotient
group (factor group) [NB: material from beginning of chap 9.]
