Abstract Algebra (Part 1)

Problem 1

How many elements of (Subscript[D, 500]) have order 500?

Problem 2

If α=(1,2,3,4)(5,6,7,1,8),what is (α^407)?

Problem 3

If G is the group of all rotations of the a cube, what is |G|?

Problem 4

How many group homomorphisms are there from (Subscript[ℤ, 15]) to (Subscript[D, 77])?

Problem 5

If G has the following Cayley table, what  must G be isomorphic to?

p5a=Join[{0}, RandomSample[ Range[8],8]];
headings= Map[ #-> (Subscript["a", (Position[p5a,#][[1,1]])])&, p5a];
table= Style[TableForm[ Outer[ Mod[#1+#2,9]&,p5a,p5a] /.headings, TableHeadings-> ( {p5a,p5a}/. headings)], FontSize-> 16, FontFamily->"Times",Italic]

Problem 6

How many elements of (Subscript[S, 6]) have order 4?

Problem 7

If G is a group of order 35, how many subgroups of order 5 does G have?  How many subgroups of order 7?  Find a complete list (up to to isomorphism) of what G can be isomorphic to.

Problem 8

Up to isomorphism, how many Abelian groups of order 392 are there?

Problem 9

Suppose n∈ℕ and up to isomorphism there are exactly 5 Abelian groups of order n.  What does this tell you about n?