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January 10th, 2008, 12:17 PM   #1
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[2 problems] Number of abelian groups and density of primes

Hi there. I have two minor problems I would like to verify. If someone could please send me clues about the below problems I would be grateful.

1. How many different (up to isomorphism) abelian groups of cardinality 245 exist?

It is just how to start problem, and why is it 245?

2. Calculate the density of primes in the arithmetic progression 10n + 3.

Completely no idea how to start. I probably lack some important theorems.

Thank You for clues.
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January 14th, 2008, 01:14 PM   #2
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Re: [2 problems] Number of abelian groups and density of pri

Quote:
Originally Posted by Creebo
Hi there. I have two minor problems I would like to verify. If someone could please send me clues about the below problems I would be grateful.

1. How many different (up to isomorphism) abelian groups of cardinality 245 exist?

It is just how to start problem, and why is it 245?

2. Calculate the density of primes in the arithmetic progression 10n + 3.

Completely no idea how to start. I probably lack some important theorems.

Thank You for clues.
1. Cheaty answer: see this list for A000688.

The important thing to note is that this is a multiplicative function, so the number of a(245) of abelian groups of order 245 = a(5) * a(7^2), and a(p^n) = n for p prime.

2. the theorem you need is Dirichlet's theorem.
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