The given problem:
How many different non-isomorphic Abelian groups of order 4 are there?
The solution given was
The number of Abelian groups of order $p^k$ ($p$ is prime) is the number of partitions of $k$. Here order is 4 i.e. 22. Partitions of 2 are {1,1},{2,0}. Total 2 partition, so number of different abelian groups are 2.
I really don't get from where solution comes and what it means. Reading online, I came to know that this is based on Sylow theorems. I learnt Sylow theorems. I guess only first theorem should makes sense in the context of this problem (or am I wrong and other Sylow theorems forms basis for the above solution). The first Sylow theorem is :
Let $G$ be a finite group, $p$ a prime and $p^r$ the highest power of $p$ dividing the order of $G$. Then there is a subgroup of $G$ of order $p^r$.
So Sylow's theorems talks about what order of subgroup always exists for given group by putting restriction on orders (unlike Lagrange's theorem which goes reverse: talks about relation between orders of group and its subgroup assuming existence of subgroup for given group).
If you compare this theorem statement and solution, you will realize that solution does not at all talk about subgroups. Also I am not able to connect non isomorphism with Sylow theorem. Also I dont understand what those groupings / partitions given in solution means. Its not that I dont understand what are partitions, but its that how particular partition maps to specific instance of Abelian group. For example, how {1,1} maps to particular abelian group, because solution says there are two partitions, hence two abelian groups. So there has to be a mapping from partition to abelian group. But I am not able to get it.
(PS: I also understand these definitions:
- Suppose is a subgroup of of order a power of a prime , and is the highest power of that divides . Then is called a Sylow p-group of G.
- In general a group of order a power of the prime is called a -group.
If they help in explaining in answers. )