I'm in the latter stages of my third year (of four) studying maths at university. I have to pick my modules for my fourth year, and we have the opportunity to do a project for one of our modules. This normally entails independently reading around a subject and writing a 30-40 page document that...
Homework Statement
Let x be the number of successes in n independent Bernoulli trials, each one having unknown probability θ of success. Assume θ has prior distribution θ ~ Unif(0,1). An extra trial, z, is performed, independent of the first n given θ, but with probability θ/2 of success. Show...
Sorry, I'm going to have to ask for a hint as to where to go from there. My ideas for proofs tend to break down when I want to assume a_{i} \in H for some a_{i}when G = \mathbb{Z}a_{1} + ... + \mathbb{Z}a_{n}.
Does this work?
Let K \le H be finitely generated where H/K is finitely generated, say K = \mathbb{Z}a_{1} + ... + \mathbb{Z}a_{n} and H/K = \mathbb{Z}(b_{1} + K) + ... + \mathbb{Z}(b_{m} + K) = \mathbb{Z}b_{1} + ... + \mathbb{Z}b_{m} + K, where the b_{i} \in H. Let h \in H, and consider h +...
Homework Statement
Prove that any subgroup of a finitely generated abelian group is finitely generated.
Homework Equations
The Attempt at a Solution
I've attempted a proof by induction on the number of generators. The case n=1 corresponds to a cyclic group, and any subgroup of a...