6. Prove that the group Q (equipped with +) is not cyclic. Hint: Suppose me Q....
suppose f; Q→Z5. Prove that the kernel of f has to be all of Q. Hint { if I subset of F is non zero ideal the I=F. what does this tell us about the ideals of fields?}
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor dofn there is a subgroup of Ghaving order d. (b) Characterize all factor groups of Z70.
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70 -
3 Let p and q be prime numbers and let G be a non-cyclic group of order pq. Let H be a subgroup of G.Show that either H is cyclic or H-G. 12 - Let I and J, be ideals in R. In, the homomorphismJ f: (!+J a → a+J use the First Isomorphism Theorem to prove that I+J
2. Suppose that < a> is a cyclic group of order 10. Find all the generators in terms of a)
Suppose we tried to apply our real analysis definitions/methods
to the
set of rational numbers Q. In other words, in the definitions, we
only
consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc.
In
this setting:
(a) Find an open cover of [0, 1] that contains no finite subcover.
Hint:
Fix an irrational number α ∈ [0, 1] (as a subset of the reals
now!)
and for each (rational) q ∈ [0, 1] look for an...
Abstract Algebra 1 a) Prove that if G is a cyclic group of prime order than G has exactly two subgroups. What are they? 1 b) Let G be a group and H a subgroup of G. Let x ∈ G. Proof that if for a, b ∈ H and ax = b then x ∈ H. (If you use any group axioms, show them)
Problem 1. Let A event from outcome space S,equipped with probability function I . Prove that P(A) 1. Hint: You can use theorem 1.4 Problem 2. Let A,B,C events from outcome space S, equipped with probability function P. Prove that P(AUBUC)- P(A)+P(B)+P(C)- PAnB) PAnC) PBnC) +P(AnBnc) Hint: You can treat A and BUC as two events and apply theorem 1.6. You will also need to use Law 5 from the distributive laws.
Problem 1. Let A event from outcome space S,equipped with probability function I . Prove that P(A) 1. Hint: You can use theorem 1.4 Problem 2. Let A,B,C events from outcome space S, equipped with probability function P. Prove that P(AUBUC)- P(A)+P(B)+P(C)- PAnB) PAnC) PBnC) +P(AnBnc) Hint: You can treat A and BUC as two events and apply theorem 1.6. You will also need to use Law 5 from the distributive laws.
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q. In other words, in the definitions, we only consider rational numbers. E.g., [0, 1] now means [0, 1] n Q, etc. In this setting: (a) Find an open cover of [0, 1] that contains no finite subcover. Hint: Fix an irrational number a € [0, 1] (as a subset of the reals now!) and for each (rational) qe [0, 1] look for an open...