Math 429 Test #1 Provide all work in the space provided! Name___________________
1. The set of integers with the operation ( ) 11 a b a b * = + – , where ,a b Z ? , satisfies the conditions needed to define a group. Find and verify the identity element and the inverse of aZ ? ,with respect to this operation. Is it an abelian group?
2. Using this exact wording and only this one sentence, finish the following. a) If f is surjective and :f C D ? , then each element of …..
b) If f is injective and :f C D ? , then each element of ……
3. Define what we mean by isomorphic binary structures.
4. Prove that the identity element of any group is unique.
5. For the following mapping, determine if the mapping is one to one and if it is onto. Justify all your answers with algebraic evidence and/or algebraically justify all counter-examples. 35 5 () x x fx – + = where : \{ 5} f R R -?
6. For any group G, prove that for any 1 1 1 *, ,( * ) a b G a b b a – – – ?= .
7. Determine whether the following binary operations give a group structure on the given set. If not, give a reason they fail and supply a counter-example. You do not need to prove or show algebraic evidence for the properties that are satisfied.
a) The set of all positive rational numbers with the operation of multiplication.
b) The set of all even integers with the operation of multiplication.
c) The rational numbers under multiplication.
d) The set of all complex numbers z that have an absolute value of 1 with multiplication.
e) The set of all complex numbers z that have an absolute value of 1 with addition.
f) For a fixed positive integer n, the set of all complex numbers 1 nx = with multiplication.
8. In the following, a relation R is defined on set of all human beings. Determine whether or not each relation is reflexive, symmetric, and/or transitive.
a) xRy if and only if x lives within 400 miles of y.
b) xRy if and only if x is the father of y.
c) xRy if and only if x is the first cousin of y.
9. Suppose Ga = is a cyclic group of order 12. List all distinct generators of G and list all distinct subgroups of G.
10. Determine whether the following mappings are isomorphisms of the first binary structure with the second. If not, explain why and give a counter-example. a) 2 , , ( ) Q with Q by x x ? ••= for xQ ?
b) , , ( ) 2 xR with R by x ? + + • = for xR ?
11. Find the order of the cyclic subgroup 8 U generated by 33 cos sin 22 i pp + . Show work.
12. Let { } { } 2 : , ( ) , 2, 1,0,1,2 , 2,4,7,11 f Z Z such that f x x S and T ? = = – – = . If they exist, find the following.
a) () fS=
b) 1() fT – =
13. Give the definition of a function and clearly specify what characteristics would allow it to be injective and/or surjective.
14. Give a complete definition of a group.
