Download Z n

Algebra (I)
Test 1
2013, 10, 18
1. Prove or disprove the following statements. (2 points each)
(a) Let Zn* = Zn \ {0}. Then < Z13*, ·> is a group.
(b) Every group with 6 elements is abelian.
(c) Let G = R \ {-1} and x。y = x + y + x·y for all x and y in G.
Then <G,。> is a group. (R is the set of all real numbers.)
(d) Any two groups of order 4 (with 4 elements) are
isomorphic.
(e) Let G be the set of all invertible 22 real matrices. Then
<G, ·> is group which is not abelian.
2. Prove that every group with at most 5 elements is abelian. (5
points)
3. (Bonus) Let (n) denote the number of positive integers not
greater than n which are relatively prime with n. Prove that
< Zn, +> is a cyclic group which has (n) generators. (3 points)