Problem 1.
Find all groups of order 6.
Problem 2.
Prove that the elements AB
and BA are of the same
order.
Problem 3.
If A is the operation
that replaces x by ax + b, prove that A is of finite
order if, and only if, a is root of unity other than 1.
Problem 4.
Show that a group of even order contains an odd number of elements
of order 2.
Problem 5.
Prove that if a finite set of matrices forms a group, the eigenvalues of
each matrix are roots of unity.
Problem 6.
Prove that those elements of a group G which are mapped onto the unit
matrix of a representation of the group G form an invariant subgroup and
that every set of elements which are mapped onto the same matrix of the
representation is a coset of this subgroup. Also prove that all
elements of
a coset are mapped onto the same matrix.
Problem 7.
The geometrical symmetry operations of a two-dimensional square
form a group. Find the order of that group, its subgroups, its class
structure,
its irreducible representations, and its character table.