- If and are finite cyclic
groups such that
and
is defined as in the
text. Show is 1-1, onto, and operation preserving.
- Let
with .
Prove that has a unique subgroup of order where .
*HINT: Let then is a subgroup of order . Suppose is another subgroup of order . Show .* - Let
with .
Let be integers such that ().
Suppose
and
.
Prove that
and
.
*HINT: For the latter, use the fact that the gcd is a linear combination of and , i.e., Theorem 1.2.3.)* - Let be a group and such that where
. Show that one can write
where , , and .
Moreover, prove the uniqueness of such a representation.
*HINT: Write , let , and use Theorem 5.2.2. For uniqueness, show using , that if and satisfy the stated conditions in the problem, then they must be given as stated in this hint.* - Prove that a group of order , where is a prime
and
, must contain a subgroup of
order .
*HINT: Use Theorem 5.2.1.* - If in a group of order , for each positive ,
the equation has less than
solutions, then show is cyclic.
*HINT: Use the Corollary 5.2.3 and Theorem 5.2.4.* (W. Wardlaw) Prove that a finite group is cyclic if and only if has no more than roots of for every , where is the identity of . .

*HINT: A root of is a solution to . Use Theorem 5.2.1 for the only if part and the Corollary 5.2.5 for the if part.*

David Joyner 2007-08-06