Algebra - Math 320 -- Fall 2009

 14.4: Which of the following is the intended interpretation of problem 14.4 in the book?

1. Show that for every abelian group G, |G| is not divisible by the square of a prime iff G is cyclic.
2. Show that n is not divisible by the square of a prime iff every abelian group of order n is cyclic.

14.5: Answer all of the following questions:

• Where it says: "... there are essentially only five groups of order 8 ..."   What does "essentially" mean here?
• Why does bab^(-1) have to be in <a>?
• Why does bab^(-1) not equal e, a, a^2 ?
• Explain why "multiplication in G is completely determined once we know ord(b)."
• Why is ord(b) = 2 or 4 ?
• Explain why if ord(b) = 2 then G is isomorphic to D_4.
• Explain why if ord(b) = 4 then G is isomorphic to Q_8