Math 366–001 HW 2, Spring 2015
This assignment is due Monday, February 9. Feel free to work together, but be sure to
write up your own solutions. As for writing it up, please write legibly on your own paper,
including as much justification as seems necessary to get the point across.
1. Let a, b, s and t be integers. If a mod st = b mod st, show that both
a mod s = b mod s and a mod t = b mod t. (Recall that A mod N = B mod N
means that N |A − B.)
2. Use induction to prove that 2n 32n − 1 is always divisible by 17.
3. Write out the complete Cayley (multiplication) table for the dihedral group of order 6, called D3 , built from the symmetries of an equilateral triangle ∆, using labels
R0 , R120 , R240 , V, D1 and D2 in that order, where R120 is a clockwise rotation by 120
degrees and D1 is the flip along the angle bisector through the bottom left angle.
4. Is D3 Abelian? Why or why not?
5. Give two reasons why the set of odd integers under addition is not a group.
6. The set of integers under subtraction is not a group, at least in part because subtraction
is not associative. Give an example of why this is true.
7. Show that {1, 2, 3} with multiplication mod 4 is not a group.
8. Show that {1, 2, 3, 4} with multiplication mod 5 is a group.
9. {5, 15, 25, 35} is a group under multiplication mod 40. What is the identity element?
10. Given elements a1 , . . . , an of some group, what is the inverse of the element a1 · . . . · an ?
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