# Number of groups of a certain order given: 1) finitely generated abelian, 2) subgroups, 3) not necessarily finitely generated

1) How many finite abelian groups are there of order $1000$?

Well via the fundemental theorem of finitely generated abelian groups, we look at the factorisation for $1000$.

$1000=2^3*5^3$ and there are $3$ ways to write $2^3$ $\quad1)\quad 2\times 4$, $\quad2)\quad 2\times 2\times 2$, $\quad3) \quad8$

and $3$ ways to write $5^3$.

So there are $9$ finite abelian groups of order $1000$.

2) How many subgroups are there of a finitely generated abelian group of order $10000$?

3) What are all of the abelian groups of order $100$? These will come from $2^2\times 3^2$ so there are four:

$$\Bbb Z_2\times \Bbb Z_2 \times \Bbb Z_5\times \Bbb Z_5$$ $$\Bbb Z_2\times \Bbb Z_2\times \Bbb Z_{25}$$ $$\Bbb Z_4\times \Bbb Z_5\times \Bbb Z_5$$ $$\Bbb Z_4\times \Bbb Z_{25}$$

Or there are $5$ with $\Bbb Z_{100}$

Firstly have I done $1)$&$3)$ right? Secondly, is $2)$ worked out via the same method?

• on Qn 3.) A group of size 100 is finitely generated, e.g, Excluding the identity the 99 elements from there will definitely generate the group. – P Vanchinathan Jun 21 '15 at 3:53
• @angryavian Should be fixed now, so sorry – Permute Jun 21 '15 at 4:14
• #1 and #3 look correct. #2 is still ambiguous: you need to know which abelian group of order 10000 the question is referring to... – angryavian Jun 21 '15 at 4:17
• All groups of order $10000$ are finitely generated so why write "finitely generated group of order $10000$"? – Derek Holt Jun 21 '15 at 11:00
• Question 2 seems a bit pointless and something you would just calculate on a computer. For example, $C_{10}^4$ seems to have $75040$ subgroups. – Derek Holt Jun 21 '15 at 11:36