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.